3.606 \(\int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=444 \[ \frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}} \]

[Out]

-1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(7/2)/(a^2+b^2)^3/d-1/2*(a-
b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(
1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/
d*2^(1/2)+1/4*(-8*a^4-31*a^2*b^2-15*b^4)/a^3/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)+1/2*b^2/a/(a^2+b^2)/d/tan(d*x+c)^(
1/2)/(a+b*tan(d*x+c))^2+1/4*b^2*(13*a^2+5*b^2)/a^2/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))

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Rubi [A]  time = 1.10, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {b^{5/2} \left (46 a^2 b^2+63 a^4+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}-\frac {31 a^2 b^2+8 a^4+15 b^4}{4 a^3 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2
+ 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(5/2)*(63*a^4 + 46*a^2*
b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*
b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a
*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (8*a^4 + 31*a^2*b^
2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]) + b^2/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[
c + d*x])^2) + (b^2*(13*a^2 + 5*b^2))/(4*a^2*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx &=\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+5 b^2\right )-2 a b \tan (c+d x)+\frac {5}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+31 a^2 b^2+15 b^4\right )-4 a^3 b \tan (c+d x)+\frac {3}{4} b^2 \left (13 a^2+5 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\int \frac {\frac {1}{8} b \left (24 a^4+31 a^2 b^2+15 b^4\right )+a^3 \left (a^2-b^2\right ) \tan (c+d x)+\frac {1}{8} b \left (8 a^4+31 a^2 b^2+15 b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\int \frac {a^3 b \left (3 a^2-b^2\right )+a^4 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 a^3 \left (a^2+b^2\right )^3}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \operatorname {Subst}\left (\int \frac {a^3 b \left (3 a^2-b^2\right )+a^4 \left (a^2-3 b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}\\ &=-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 4.17, size = 358, normalized size = 0.81 \[ \frac {\frac {13 a^2 b^2+5 b^4}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {-46 a^{5/2} b^4-39 a^{9/2} b^2-4 (-1)^{3/4} a^{7/2} (a+i b)^3 \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}-4 \sqrt [4]{-1} a^{7/2} (b+i a)^3 \sqrt {\tan (c+d x)} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-8 a^{13/2}-63 a^4 b^{5/2} \sqrt {\tan (c+d x)} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-46 a^2 b^{9/2} \sqrt {\tan (c+d x)} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-15 b^{13/2} \sqrt {\tan (c+d x)} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-15 \sqrt {a} b^6}{a^{5/2} \left (a^2+b^2\right )^2}+\frac {2 b^2}{(a+b \tan (c+d x))^2}}{4 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

((-8*a^(13/2) - 39*a^(9/2)*b^2 - 46*a^(5/2)*b^4 - 15*Sqrt[a]*b^6 - 4*(-1)^(3/4)*a^(7/2)*(a + I*b)^3*ArcTan[(-1
)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 63*a^4*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*S
qrt[Tan[c + d*x]] - 46*a^2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 15*b^(13/
2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 4*(-1)^(1/4)*a^(7/2)*(I*a + b)^3*ArcTanh[
(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*(a^2 + b^2)^2) + (2*b^2)/(a + b*Tan[c + d*x])^2 +
(13*a^2*b^2 + 5*b^4)/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)^3*tan(d*x + c)^(3/2)), x)

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maple [B]  time = 0.31, size = 920, normalized size = 2.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x)

[Out]

-15/4/d*a/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*b^4-11/2/d*b^6/(a^2+b^2)^3/(a+b*tan(d*x+c))^2/a*tan(
d*x+c)^(3/2)-7/4/d*b^8/(a^2+b^2)^3/a^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)-17/4/d*a^2/(a^2+b^2)^3/(a+b*tan(d*x
+c))^2*b^3*tan(d*x+c)^(1/2)-13/2/d*b^5/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)-9/4/d*b^7/(a^2+b^2)^3/a
^2/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)-63/4/d*a/(a^2+b^2)^3*b^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1
/2))-23/2/d*b^5/(a^2+b^2)^3/a/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))-15/4/d*b^7/(a^2+b^2)^3/a^3/(a
*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))-2/d/a^3/tan(d*x+c)^(1/2)-3/2/d/(a^2+b^2)^3*2^(1/2)*arctan(-1+
2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3-3/4/d/(a^2+b
^2)^3*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^2*b+1/4/d/
(a^2+b^2)^3*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*b^3-3/
2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2^(1/2)*ta
n(d*x+c)^(1/2))*b^3-1/4/d/(a^2+b^2)^3*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)
^(1/2)+tan(d*x+c)))*a^3+3/4/d/(a^2+b^2)^3*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*
x+c)^(1/2)+tan(d*x+c)))*a*b^2-1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+3/2/d/(a^2+b^2
)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1
/2))*a^3+3/2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2

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maxima [A]  time = 0.78, size = 464, normalized size = 1.05 \[ -\frac {\frac {{\left (63 \, a^{4} b^{3} + 46 \, a^{2} b^{5} + 15 \, b^{7}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} + \frac {8 \, a^{6} + 16 \, a^{4} b^{2} + 8 \, a^{2} b^{4} + {\left (8 \, a^{4} b^{2} + 31 \, a^{2} b^{4} + 15 \, b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (16 \, a^{5} b + 49 \, a^{3} b^{3} + 25 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {\tan \left (d x + c\right )}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*((63*a^4*b^3 + 46*a^2*b^5 + 15*b^7)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^9 + 3*a^7*b^2 + 3*a^5*b^4
+ a^3*b^6)*sqrt(a*b)) + (8*a^6 + 16*a^4*b^2 + 8*a^2*b^4 + (8*a^4*b^2 + 31*a^2*b^4 + 15*b^6)*tan(d*x + c)^2 + (
16*a^5*b + 49*a^3*b^3 + 25*a*b^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^(5/2) + 2*(a^8*b
+ 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^(3/2) + (a^9 + 2*a^7*b^2 + a^5*b^4)*sqrt(tan(d*x + c))) + (2*sqrt(2)*(a^3
+ 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^3 + 3*a^2*b -
3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)
*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)*sqr
t(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6))/d

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mupad [B]  time = 13.71, size = 16740, normalized size = 37.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3),x)

[Out]

(log(29491200*a^22*b^35*d^4 - ((((-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i -
20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(((((-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^
4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(251658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 483
68713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^
8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 35469618315264*a^40*b^29*d^8 + 4789690485964
8*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 +
20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a^54*b^15*d^8 + 565576728576*a^56*
b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b
^5*d^8 - 167772160*a^66*b^3*d^8 + (tan(c + d*x)^(1/2)*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2
- a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*
d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37
*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 56358023
98720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9
- 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59
*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^6
7*b^5*d^9 - 134217728*a^69*b^3*d^9))/2))/2 - tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^4
2*d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a
^32*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 92
434029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46
*b^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 805425
905664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160
*a^62*b^4*d^7 + 16777216*a^64*b^2*d^7))*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2
*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - 117964800*a^21*b^42*d^6 - 841482240*a^23*b^40*d^6 + 38293
99552*a^25*b^38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162944*a^29*b^34*d^6 + 1899895980032*a^31*b^32*d^6 + 4
972695519232*a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 + 12890720436224*a^37*b^26*d^6 + 12726089809920*a^39*
b^24*d^6 + 8366961197056*a^41*b^22*d^6 + 2597662490624*a^43*b^20*d^6 - 1171836108800*a^45*b^18*d^6 - 198688165
0688*a^47*b^16*d^6 - 1237583921152*a^49*b^14*d^6 - 449507753984*a^51*b^12*d^6 - 97476149248*a^53*b^10*d^6 - 11
931222016*a^55*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 134217728*a^59*b^4*d^6 - 8388608*a^61*b^2*d^6))/2 + tan(c +
d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^3
9*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a
^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 18074
74491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 -
4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5))*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*
b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 + 460062720*a^24*b^33*d^
4 + 3439722496*a^26*b^31*d^4 + 16227237888*a^28*b^29*d^4 + 53669396480*a^30*b^27*d^4 + 131031367680*a^32*b^25*
d^4 + 242529730560*a^34*b^23*d^4 + 344454070272*a^36*b^21*d^4 + 375993532416*a^38*b^19*d^4 + 313043189760*a^40
*b^17*d^4 + 195253370880*a^42*b^15*d^4 + 88318935040*a^44*b^13*d^4 + 27352498176*a^46*b^11*d^4 + 5187043328*a^
48*b^9*d^4 + 454164480*a^50*b^7*d^4)*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15
i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - log(29491200*a^22*b^35*d^4 - ((-1/(4*(b^6*d^2*1i - a^6*d^2*1
i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(((-1/(4*(b^6*d^2*
1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(251
658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 126
8458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^
31*d^8 + 35469618315264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904
859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d
^8 + 2994733056000*a^54*b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b
^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 - tan(c + d*x)^(1/2)*(-1/
(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))
)^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^
39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 520227913728
0*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 22
54320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^1
7*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*
a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9)) + tan(c + d*x)^(1
/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^42*d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d
^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a^32*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600
*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 92434029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 +
86342935511040*a^44*b^22*d^7 + 59767095558144*a^46*b^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^
50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 805425905664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 161270988
8*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160*a^62*b^4*d^7 + 16777216*a^64*b^2*d^7))*(-1/(4*(b^6*d^2*1i -
a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) - 117964
800*a^21*b^42*d^6 - 841482240*a^23*b^40*d^6 + 3829399552*a^25*b^38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162
944*a^29*b^34*d^6 + 1899895980032*a^31*b^32*d^6 + 4972695519232*a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 +
12890720436224*a^37*b^26*d^6 + 12726089809920*a^39*b^24*d^6 + 8366961197056*a^41*b^22*d^6 + 2597662490624*a^43
*b^20*d^6 - 1171836108800*a^45*b^18*d^6 - 1986881650688*a^47*b^16*d^6 - 1237583921152*a^49*b^14*d^6 - 44950775
3984*a^51*b^12*d^6 - 97476149248*a^53*b^10*d^6 - 11931222016*a^55*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 13421772
8*a^59*b^4*d^6 - 8388608*a^61*b^2*d^6) - tan(c + d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^
5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5
+ 1104303620096*a^33*b^27*d^5 + 2240523796480*a^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 3717287903232*a^3
9*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 1807474491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 17076899
0208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 - 4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^
55*b^5*d^5))*(-1/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 +
a^4*b^2*d^2*15i)))^(1/2) + 460062720*a^24*b^33*d^4 + 3439722496*a^26*b^31*d^4 + 16227237888*a^28*b^29*d^4 + 53
669396480*a^30*b^27*d^4 + 131031367680*a^32*b^25*d^4 + 242529730560*a^34*b^23*d^4 + 344454070272*a^36*b^21*d^4
+ 375993532416*a^38*b^19*d^4 + 313043189760*a^40*b^17*d^4 + 195253370880*a^42*b^15*d^4 + 88318935040*a^44*b^1
3*d^4 + 27352498176*a^46*b^11*d^4 + 5187043328*a^48*b^9*d^4 + 454164480*a^50*b^7*d^4)*(-1/(4*(b^6*d^2*1i - a^6
*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) - (2/a + (ta
n(c + d*x)^2*(15*b^6 + 31*a^2*b^4 + 8*a^4*b^2))/(4*a^3*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(16*a^4*b + 25
*b^5 + 49*a^2*b^3))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d*tan(c + d*x)^(1/2) + b^2*d*tan(c + d*x)^(5/2) + 2*
a*b*d*tan(c + d*x)^(3/2)) + atan((((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 -
a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2
*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(251658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 483
68713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^
8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 35469618315264*a^40*b^29*d^8 + 4789690485964
8*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 +
20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a^54*b^15*d^8 + 565576728576*a^56*
b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b
^5*d^8 - 167772160*a^66*b^3*d^8 + tan(c + d*x)^(1/2)*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i
- 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d
^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*
b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 563580239
8720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 -
6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*
b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67
*b^5*d^9 - 134217728*a^69*b^3*d^9)) - tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^42*d^7 +
91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a^32*b^3
4*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 924340296
08960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46*b^20*d
^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 805425905664*
a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160*a^62*b
^4*d^7 + 16777216*a^64*b^2*d^7))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a
^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - 117964800*a^21*b^42*d^6 - 841482240*a^23*b^40*d^6 + 3829399552*a^25
*b^38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162944*a^29*b^34*d^6 + 1899895980032*a^31*b^32*d^6 + 49726955192
32*a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 + 12890720436224*a^37*b^26*d^6 + 12726089809920*a^39*b^24*d^6 +
8366961197056*a^41*b^22*d^6 + 2597662490624*a^43*b^20*d^6 - 1171836108800*a^45*b^18*d^6 - 1986881650688*a^47*
b^16*d^6 - 1237583921152*a^49*b^14*d^6 - 449507753984*a^51*b^12*d^6 - 97476149248*a^53*b^10*d^6 - 11931222016*
a^55*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 134217728*a^59*b^4*d^6 - 8388608*a^61*b^2*d^6) + tan(c + d*x)^(1/2)*(
7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774
565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a^35*b^25*d^5
+ 3345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 1807474491392*a^43
*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 - 4917821440*a
^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*
b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*1i - ((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5
*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(((-1i/(4*(b^6*d^2 - a^6*d
^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(251658240*a^24*
b^45*d^8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a
^32*b^37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 354
69618315264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*
b^23*d^8 + 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733
056000*a^54*b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 160
89350144*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 - tan(c + d*x)^(1/2)*(-1i/(4*(b^6*d^2
- a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(1342177
28*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 4972766
82240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9
+ 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49
*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 31213674
82368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 2
2817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9)) + tan(c + d*x)^(1/2)*(471859200*a
^22*b^44*d^7 + 9500098560*a^24*b^42*d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 24646485278
72*a^30*b^36*d^7 + 8104469069824*a^32*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 +
69534945902592*a^38*b^28*d^7 + 92434029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a
^44*b^22*d^7 + 59767095558144*a^46*b^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 40
30457708544*a^52*b^14*d^7 + 805425905664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 +
16777216*a^60*b^6*d^7 + 167772160*a^62*b^4*d^7 + 16777216*a^64*b^2*d^7))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d
^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - 117964800*a^21*b^42*d^6 -
841482240*a^23*b^40*d^6 + 3829399552*a^25*b^38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162944*a^29*b^34*d^6 +
1899895980032*a^31*b^32*d^6 + 4972695519232*a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 + 12890720436224*a^37*
b^26*d^6 + 12726089809920*a^39*b^24*d^6 + 8366961197056*a^41*b^22*d^6 + 2597662490624*a^43*b^20*d^6 - 11718361
08800*a^45*b^18*d^6 - 1986881650688*a^47*b^16*d^6 - 1237583921152*a^49*b^14*d^6 - 449507753984*a^51*b^12*d^6 -
97476149248*a^53*b^10*d^6 - 11931222016*a^55*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 134217728*a^59*b^4*d^6 - 838
8608*a^61*b^2*d^6) - tan(c + d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*
b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^3
3*b^27*d^5 + 2240523796480*a^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967
114240*a^41*b^19*d^5 + 1807474491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 +
10492051456*a^49*b^11*d^5 - 4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5))*(-1i/(4
*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)
*1i)/(58982400*a^22*b^35*d^4 - ((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^
3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^
4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(251658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 483687
13728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^8 +
10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 35469618315264*a^40*b^29*d^8 + 47896904859648*a
^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 + 20
487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a^54*b^15*d^8 + 565576728576*a^56*b^1
3*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b^5*
d^8 - 167772160*a^66*b^3*d^8 - tan(c + d*x)^(1/2)*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 1
5*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9
+ 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^3
5*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 563580239872
0*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 65
02848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^1
3*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^
5*d^9 - 134217728*a^69*b^3*d^9)) + tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^42*d^7 + 91
857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a^32*b^34*d
^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 924340296089
60*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46*b^20*d^7
+ 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 805425905664*a^5
4*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160*a^62*b^4*
d^7 + 16777216*a^64*b^2*d^7))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*
b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - 117964800*a^21*b^42*d^6 - 841482240*a^23*b^40*d^6 + 3829399552*a^25*b^
38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162944*a^29*b^34*d^6 + 1899895980032*a^31*b^32*d^6 + 4972695519232*
a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 + 12890720436224*a^37*b^26*d^6 + 12726089809920*a^39*b^24*d^6 + 83
66961197056*a^41*b^22*d^6 + 2597662490624*a^43*b^20*d^6 - 1171836108800*a^45*b^18*d^6 - 1986881650688*a^47*b^1
6*d^6 - 1237583921152*a^49*b^14*d^6 - 449507753984*a^51*b^12*d^6 - 97476149248*a^53*b^10*d^6 - 11931222016*a^5
5*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 134217728*a^59*b^4*d^6 - 8388608*a^61*b^2*d^6) - tan(c + d*x)^(1/2)*(761
0564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774565
376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a^35*b^25*d^5 + 3
345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 1807474491392*a^43*b^
17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 - 4917821440*a^51
*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d
^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - ((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6
i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(((-1i/(4*(b^6*d^2 - a^6*d^2 + a
*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(251658240*a^24*b^45*d
^8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^
37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 354696183
15264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d
^8 + 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000
*a^54*b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 160893501
44*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 + tan(c + d*x)^(1/2)*(-1i/(4*(b^6*d^2 - a^6
*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(134217728*a^2
7*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*
a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502
848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*
d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*
a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 2281701
3760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9)) - tan(c + d*x)^(1/2)*(471859200*a^22*b^
44*d^7 + 9500098560*a^24*b^42*d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^3
0*b^36*d^7 + 8104469069824*a^32*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534
945902592*a^38*b^28*d^7 + 92434029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^
22*d^7 + 59767095558144*a^46*b^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 40304577
08544*a^52*b^14*d^7 + 805425905664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777
216*a^60*b^6*d^7 + 167772160*a^62*b^4*d^7 + 16777216*a^64*b^2*d^7))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i
+ a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - 117964800*a^21*b^42*d^6 - 841482
240*a^23*b^40*d^6 + 3829399552*a^25*b^38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162944*a^29*b^34*d^6 + 189989
5980032*a^31*b^32*d^6 + 4972695519232*a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 + 12890720436224*a^37*b^26*d
^6 + 12726089809920*a^39*b^24*d^6 + 8366961197056*a^41*b^22*d^6 + 2597662490624*a^43*b^20*d^6 - 1171836108800*
a^45*b^18*d^6 - 1986881650688*a^47*b^16*d^6 - 1237583921152*a^49*b^14*d^6 - 449507753984*a^51*b^12*d^6 - 97476
149248*a^53*b^10*d^6 - 11931222016*a^55*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 134217728*a^59*b^4*d^6 - 8388608*a
^61*b^2*d^6) + tan(c + d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d
^5 - 58982400*a^21*b^39*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27
*d^5 + 2240523796480*a^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240
*a^41*b^19*d^5 + 1807474491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492
051456*a^49*b^11*d^5 - 4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5))*(-1i/(4*(b^6*
d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) + 920
125440*a^24*b^33*d^4 + 6879444992*a^26*b^31*d^4 + 32454475776*a^28*b^29*d^4 + 107338792960*a^30*b^27*d^4 + 262
062735360*a^32*b^25*d^4 + 485059461120*a^34*b^23*d^4 + 688908140544*a^36*b^21*d^4 + 751987064832*a^38*b^19*d^4
+ 626086379520*a^40*b^17*d^4 + 390506741760*a^42*b^15*d^4 + 176637870080*a^44*b^13*d^4 + 54704996352*a^46*b^1
1*d^4 + 10374086656*a^48*b^9*d^4 + 908328960*a^50*b^7*d^4))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*
d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*2i + (atan((((tan(c + d*x)^(1/2)*(76105646
08*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774565376*a
^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a^35*b^25*d^5 + 334524
9468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 1807474491392*a^43*b^17*d^
5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 - 4917821440*a^51*b^9*
d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5) + ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(11796
4800*a^21*b^42*d^6 + 841482240*a^23*b^40*d^6 - 3829399552*a^25*b^38*d^6 - 78068580352*a^27*b^36*d^6 - 49743816
2944*a^29*b^34*d^6 - 1899895980032*a^31*b^32*d^6 - 4972695519232*a^33*b^30*d^6 - 9371195015168*a^35*b^28*d^6 -
12890720436224*a^37*b^26*d^6 - 12726089809920*a^39*b^24*d^6 - 8366961197056*a^41*b^22*d^6 - 2597662490624*a^4
3*b^20*d^6 + 1171836108800*a^45*b^18*d^6 + 1986881650688*a^47*b^16*d^6 + 1237583921152*a^49*b^14*d^6 + 4495077
53984*a^51*b^12*d^6 + 97476149248*a^53*b^10*d^6 + 11931222016*a^55*b^8*d^6 + 1006632960*a^57*b^6*d^6 + 1342177
28*a^59*b^4*d^6 + 8388608*a^61*b^2*d^6 - ((tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^42*
d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a^3
2*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 9243
4029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46*b
^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 80542590
5664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160*a
^62*b^4*d^7 + 16777216*a^64*b^2*d^7) + ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(251658240*a^24*b^45*d
^8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^
37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 354696183
15264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d
^8 + 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000
*a^54*b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 160893501
44*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 - (tan(c + d*x)^(1/2)*(-a^7*b^5)^(1/2)*(63*
a^4 + 15*b^4 + 46*a^2*b^2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 1
27506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33
*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488
*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 520
2279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*
d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9)
)/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)
))*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))/(8
*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)))*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*1i)/(8*(a
^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)) + ((tan(c + d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*
a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a
^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 37172
87903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 1807474491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^
5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 - 4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5
+ 8388608*a^55*b^5*d^5) - ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(117964800*a^21*b^42*d^6 + 84148224
0*a^23*b^40*d^6 - 3829399552*a^25*b^38*d^6 - 78068580352*a^27*b^36*d^6 - 497438162944*a^29*b^34*d^6 - 18998959
80032*a^31*b^32*d^6 - 4972695519232*a^33*b^30*d^6 - 9371195015168*a^35*b^28*d^6 - 12890720436224*a^37*b^26*d^6
- 12726089809920*a^39*b^24*d^6 - 8366961197056*a^41*b^22*d^6 - 2597662490624*a^43*b^20*d^6 + 1171836108800*a^
45*b^18*d^6 + 1986881650688*a^47*b^16*d^6 + 1237583921152*a^49*b^14*d^6 + 449507753984*a^51*b^12*d^6 + 9747614
9248*a^53*b^10*d^6 + 11931222016*a^55*b^8*d^6 + 1006632960*a^57*b^6*d^6 + 134217728*a^59*b^4*d^6 + 8388608*a^6
1*b^2*d^6 + ((tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^42*d^7 + 91857354752*a^26*b^40*d
^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a^32*b^34*d^7 + 20769933361152*a
^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 92434029608960*a^40*b^26*d^7 + 99
508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46*b^20*d^7 + 32432589897728*a^48
*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 805425905664*a^54*b^12*d^7 + 86608183
296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160*a^62*b^4*d^7 + 16777216*a^64*b
^2*d^7) - ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(251658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8
+ 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b
^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 35469618315264*a^40*b^29*d^8 + 4789690
4859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21
*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a^54*b^15*d^8 + 565576728576
*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*
a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 + (tan(c + d*x)^(1/2)*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(1
34217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 4
97276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^3
1*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 225432095948
8*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 31
21367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d
^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9))/(8*(a^13*d + a^7*b^6*d + 3*
a^9*b^4*d + 3*a^11*b^2*d))))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)))*(-a^7*b^5)^(1/2)*(63*a^4 +
15*b^4 + 46*a^2*b^2))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))/(8*(a^13*d + a^7*b^6*d + 3*a^9*
b^4*d + 3*a^11*b^2*d)))*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*1i)/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4
*d + 3*a^11*b^2*d)))/(58982400*a^22*b^35*d^4 + 920125440*a^24*b^33*d^4 + 6879444992*a^26*b^31*d^4 + 3245447577
6*a^28*b^29*d^4 + 107338792960*a^30*b^27*d^4 + 262062735360*a^32*b^25*d^4 + 485059461120*a^34*b^23*d^4 + 68890
8140544*a^36*b^21*d^4 + 751987064832*a^38*b^19*d^4 + 626086379520*a^40*b^17*d^4 + 390506741760*a^42*b^15*d^4 +
176637870080*a^44*b^13*d^4 + 54704996352*a^46*b^11*d^4 + 10374086656*a^48*b^9*d^4 + 908328960*a^50*b^7*d^4 +
((tan(c + d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400
*a^21*b^39*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 224052
3796480*a^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^
5 + 1807474491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b
^11*d^5 - 4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5) + ((-a^7*b^5)^(1/2)*(63*a^4
+ 15*b^4 + 46*a^2*b^2)*(117964800*a^21*b^42*d^6 + 841482240*a^23*b^40*d^6 - 3829399552*a^25*b^38*d^6 - 780685
80352*a^27*b^36*d^6 - 497438162944*a^29*b^34*d^6 - 1899895980032*a^31*b^32*d^6 - 4972695519232*a^33*b^30*d^6 -
9371195015168*a^35*b^28*d^6 - 12890720436224*a^37*b^26*d^6 - 12726089809920*a^39*b^24*d^6 - 8366961197056*a^4
1*b^22*d^6 - 2597662490624*a^43*b^20*d^6 + 1171836108800*a^45*b^18*d^6 + 1986881650688*a^47*b^16*d^6 + 1237583
921152*a^49*b^14*d^6 + 449507753984*a^51*b^12*d^6 + 97476149248*a^53*b^10*d^6 + 11931222016*a^55*b^8*d^6 + 100
6632960*a^57*b^6*d^6 + 134217728*a^59*b^4*d^6 + 8388608*a^61*b^2*d^6 - ((tan(c + d*x)^(1/2)*(471859200*a^22*b^
44*d^7 + 9500098560*a^24*b^42*d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^3
0*b^36*d^7 + 8104469069824*a^32*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534
945902592*a^38*b^28*d^7 + 92434029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^
22*d^7 + 59767095558144*a^46*b^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 40304577
08544*a^52*b^14*d^7 + 805425905664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777
216*a^60*b^6*d^7 + 167772160*a^62*b^4*d^7 + 16777216*a^64*b^2*d^7) + ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a
^2*b^2)*(251658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^
39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 2146282399
3344*a^38*b^31*d^8 + 35469618315264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^
8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608
*a^52*b^17*d^8 + 2994733056000*a^54*b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 502813
16352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 - (tan(c + d*
x)^(1/2)*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 +
22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35
*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720
*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 650
2848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13
*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5
*d^9 - 134217728*a^69*b^3*d^9))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))/(8*(a^13*d + a^7*b^6*d
+ 3*a^9*b^4*d + 3*a^11*b^2*d)))*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2))/(8*(a^13*d + a^7*b^6*d + 3*a
^9*b^4*d + 3*a^11*b^2*d))))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)))*(-a^7*b^5)^(1/2)*(63*a^4 +
15*b^4 + 46*a^2*b^2))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)) - ((tan(c + d*x)^(1/2)*(7610564608
*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^39*d^5 + 85774565376*a^2
9*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a^35*b^25*d^5 + 33452494
68416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 1807474491392*a^43*b^17*d^5
+ 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 - 4917821440*a^51*b^9*d^
5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5) - ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(1179648
00*a^21*b^42*d^6 + 841482240*a^23*b^40*d^6 - 3829399552*a^25*b^38*d^6 - 78068580352*a^27*b^36*d^6 - 4974381629
44*a^29*b^34*d^6 - 1899895980032*a^31*b^32*d^6 - 4972695519232*a^33*b^30*d^6 - 9371195015168*a^35*b^28*d^6 - 1
2890720436224*a^37*b^26*d^6 - 12726089809920*a^39*b^24*d^6 - 8366961197056*a^41*b^22*d^6 - 2597662490624*a^43*
b^20*d^6 + 1171836108800*a^45*b^18*d^6 + 1986881650688*a^47*b^16*d^6 + 1237583921152*a^49*b^14*d^6 + 449507753
984*a^51*b^12*d^6 + 97476149248*a^53*b^10*d^6 + 11931222016*a^55*b^8*d^6 + 1006632960*a^57*b^6*d^6 + 134217728
*a^59*b^4*d^6 + 8388608*a^61*b^2*d^6 + ((tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^42*d^
7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a^32*
b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 924340
29608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46*b^2
0*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 8054259056
64*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160*a^6
2*b^4*d^7 + 16777216*a^64*b^2*d^7) - ((-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*(251658240*a^24*b^45*d^8
+ 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^37
*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 35469618315
264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d^8
+ 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a
^54*b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 16089350144
*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 + (tan(c + d*x)^(1/2)*(-a^7*b^5)^(1/2)*(63*a^
4 + 15*b^4 + 46*a^2*b^2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127
506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d
^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a
^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 52022
79137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^
9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9))/
(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)))
*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2))/(8*(a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))/(8*(
a^13*d + a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d)))*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2))/(8*(a^13*d
+ a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))))*(-a^7*b^5)^(1/2)*(63*a^4 + 15*b^4 + 46*a^2*b^2)*1i)/(4*(a^13*d +
a^7*b^6*d + 3*a^9*b^4*d + 3*a^11*b^2*d))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Integral(1/((a + b*tan(c + d*x))**3*tan(c + d*x)**(3/2)), x)

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