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

Optimal. Leaf size=396 \[ -\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d \left (a^2+b^2\right )^3} \]

[Out]

1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(5/2)/(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)+1/2*b^2*tan(d*x+c)^(1/2)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+1/4*b^2*(11*a^2+3*b^2)*tan(d*x+c)^(1/2)/a^2/(
a^2+b^2)^2/d/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.84, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 16, 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} \[ \frac {b^{3/2} \left (6 a^2 b^2+35 a^4+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Tan[c + d*x]]*(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^(3/2)*(35*a^4 + 6*a^
2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(5/2)*(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) + (b^2*Sqrt[Tan[c +
d*x]])/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b^2*(11*a^2 + 3*b^2)*Sqrt[Tan[c + d*x]])/(4*a^2*(a^2 + b
^2)^2*d*(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}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx &=\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+3 b^2\right )-2 a b \tan (c+d x)+\frac {3}{2} b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+3 a^2 b^2+3 b^4\right )-4 a^3 b \tan (c+d x)+\frac {1}{4} b^2 \left (11 a^2+3 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a^3 \left (a^2-3 b^2\right )-2 a^2 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{2 a^2 \left (a^2+b^2\right )^3}+\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 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^2 \left (a^2+b^2\right )^3}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {2 a^3 \left (a^2-3 b^2\right )-2 a^2 b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^3 d}+\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a^2 \left (a^2+b^2\right )^3 d}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (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^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a^2 \left (a^2+b^2\right )^3 d}\\ &=\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d}+\frac {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (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-\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 {\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 {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \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 {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (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^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \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 {b^2 \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 2.75, size = 235, normalized size = 0.59 \[ \frac {\frac {\left (11 a^2 b^2+3 b^4\right ) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {-4 \sqrt [4]{-1} a^{5/2} (a+i b)^3 \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-4 \sqrt [4]{-1} a^{5/2} (a-i b)^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )^2}+\frac {2 b^2 \sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2}}{4 a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(-1)^(1/4)*a^(5/2)*(a + I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^
4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 4*(-1)^(1/4)*a^(5/2)*(a - I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan
[c + d*x]]])/(a^(3/2)*(a^2 + b^2)^2) + (2*b^2*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])^2 + ((11*a^2*b^2 + 3*b^
4)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d)

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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)^(1/2)/(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 \[ \int \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.37, size = 903, normalized size = 2.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/4/d*a^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^3*tan(d*x+c)^(3/2)+7/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)
^(3/2)*b^5+3/4/d*b^7/(a^2+b^2)^3/(a+b*tan(d*x+c))^2/a^2*tan(d*x+c)^(3/2)+13/4/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+c
))^2*b^2*tan(d*x+c)^(1/2)+9/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)*a*b^4+5/4/d*b^6/(a^2+b^2)^3/(a
+b*tan(d*x+c))^2/a*tan(d*x+c)^(1/2)+35/4/d*a^2/(a^2+b^2)^3*b^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/
2))+3/2/d/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*b^4+3/4/d*b^6/(a^2+b^2)^3/a^2/(a*b)^(
1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))+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+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^2-3/2/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^3-3/2/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)*arct
an(-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

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maxima [A]  time = 0.43, size = 419, normalized size = 1.06 \[ \frac {\frac {{\left (35 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (11 \, a^{2} b^{3} + 3 \, b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (13 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((35*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a
^2*b^6)*sqrt(a*b)) + (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)))) + sqr
t(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)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) +
((11*a^2*b^3 + 3*b^5)*tan(d*x + c)^(3/2) + (13*a^3*b^2 + 5*a*b^4)*sqrt(tan(d*x + c)))/(a^8 + 2*a^6*b^2 + a^4*
b^4 + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*tan(d*x + c)))/d

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mupad [B]  time = 9.64, size = 17833, normalized size = 45.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c + d*x)^(1/2)*(5*b^4 + 13*a^2*b^2))/(4*a*(a^4 + b^4 + 2*a^2*b^2)) + (b*tan(c + d*x)^(3/2)*(3*b^4 + 11*a
^2*b^2))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d + b^2*d*tan(c + d*x)^2 + 2*a*b*d*tan(c + d*x)) - 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)*((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d
^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4
+ 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*
b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - (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)*
(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504
*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^
24*b^5*d^4 - 512*a^26*b^3*d^4))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4
+ 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) + (tan(c + d*x)^(1/2)*(72*a*b^22*d^2
+ 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 7024
0*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6
*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*
b^2*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) - (90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*
d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a^20*d^5 + a^4*b^16*d^5
+ 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 +
8*a^18*b^2*d^5)) + (tan(c + d*x)^(1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802
*a^10*b^7 - 1257*a^12*b^5))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 7
0*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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 - ((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)*((192*a^2*b
^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d
^4 + 107520*a^14*b^12*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 -
128*a^24*b^2*d^4)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8
*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (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)*(512*a^4*b^25*d^4 + 46
08*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504
*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*
b^3*d^4))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 5
6*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) - (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 +
5024*a^5*b^18*d^2 + 14272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 4768
0*a^15*b^8*d^2 + 12616*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^1
2*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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) - (90
*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d
^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*
a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (ta
n(c + d*x)^(1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*
b^5))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^
14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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)/(((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)*((192*a^2*b^24*d^4 + 1728*a^4*b^2
2*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*
d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(a^2
0*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5
+ 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - (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)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 1792
0*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080
*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(a^20*d^4 +
a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^
16*b^4*d^4 + 8*a^18*b^2*d^4)) + (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14
272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616
*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d
^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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) - (90*a*b^19*d^2 + 846*a^3*
b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^
2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10
*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) + (tan(c + d*x)^(1/2)*(18*a
^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(a^20*d^4 + a^4*
b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b
^4*d^4 + 8*a^18*b^2*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)*(((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)*((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4
+ 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4 + 76800*a^16*b^10*d^4
+ 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(a^20*d^5 + a^4*b^16*d^5 + 8*
a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^
18*b^2*d^5) + (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)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^
10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^
20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14
*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d
^4)) - (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272*a^7*b^16*d^2 + 27824*
a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616*a^17*b^6*d^2 - 64*a^21*b
^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56
*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))*(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) - (90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*
d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 3
2*a^17*b^3*d^2)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d
^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (tan(c + d*x)^(1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4
*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4
+ 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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) + (9*b^14 + 60*a^2*b^12 + 318*a^4*b^10 + 748*a^6*b^8 + 1505*a^8*b^6)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*
b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b
^2*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)*2i - atan((((((((((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*
d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*
d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 +
28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) -
(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)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 +
46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 1
7920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a
^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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
))/2 + (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272*a^7*b^16*d^2 + 27824*
a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616*a^17*b^6*d^2 - 64*a^21*b
^2*d^2))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 +
56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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))/2 - (90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^
5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*
d^2 - 32*a^17*b^3*d^2)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*
a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*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 + (tan(c + d*x)^(1/2)*(18*a^2*b
^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(2*(a^20*d^4 + a^4*b
^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^
4*d^4 + 8*a^18*b^2*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)*1i - ((((((((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264
*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4 + 76800*a^16*b^10*d^4 + 33984
*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b
^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^
2*d^5)) + (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)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b
^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b
^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*
d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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))/2 - (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272*a^7*b^16*d^
2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616*a^17*b^6*d^2 -
64*a^21*b^2*d^2))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12
*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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))/2 - (90*a*b^19*d^2 + 846*a^3*b^17*d^2
+ 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258
*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10
*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*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 - (tan(c + d*x)^(1/2)
*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(2*(a^20*d
^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 +
28*a^16*b^4*d^4 + 8*a^18*b^2*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)*1i)/((9*b^14 + 60*a^2*b^12 + 318*a^4*b^10 + 748*a^6*b^8 + 1505*a^8*
b^6)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^1
4*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((((((((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6*b^20*d
^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4 + 76800*a^16*b^10*d
^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(2*(a^20*d^5 + a^4*b^16*d^5
+ 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 +
8*a^18*b^2*d^5)) - (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)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38
400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38
400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8
*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a
^18*b^2*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))/2 + (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272*a
^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616*a^17
*b^6*d^2 - 64*a^21*b^2*d^2))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4
+ 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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))/2 - (90*a*b^19*d^2 + 846*a^
3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*
d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56
*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*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 + (tan(c +
d*x)^(1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/
(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*
b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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) + ((((((((192*a^2*b^24*d^4 + 1728*a^4*b^22*d^4 + 8320*a^6
*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4 + 76800*a^16
*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(2*(a^20*d^5 + a^4*b
^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^
4*d^5 + 8*a^18*b^2*d^5)) + (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)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d
^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d
^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4*(a^20*d^4 + a^4*b^16*
d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^
4 + 8*a^18*b^2*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))/2 - (tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 +
14272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 126
16*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b
^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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))/2 - (90*a*b^19*d^2 +
846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^
13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d
^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*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 - (t
an(c + d*x)^(1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12
*b^5))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 5
6*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*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)*1i - (atan(((((tan(c + d*x)^(1/2)*(18*a^2
*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(a^20*d^4 + a^4*b^
16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4
*d^4 + 8*a^18*b^2*d^4) - (((90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 14578*a
^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a^20*d^5 + a^4*b
^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^
4*d^5 + 8*a^18*b^2*d^5) - (((tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 + 14272*
a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 12616*a^1
7*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 +
70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) + (((192*a^2*b^24*d^4 + 1728*a^4*b^22*d
^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^12*d^4
+ 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(a^20*d
^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 +
28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - (tan(c + d*x)^(1/2)*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*(512*a^4
*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^
15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d
^4 - 512*a^26*b^3*d^4))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b
^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^
2*d^4)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*
(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*(-a^5*b^3
)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*(-a^5*b^3)^(1/2)*(
35*a^4 + 3*b^4 + 6*a^2*b^2)*1i)/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)) + (((tan(c + d*x)^(1/2)*(
18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(a^20*d^4 +
a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^
16*b^4*d^4 + 8*a^18*b^2*d^4) + (((90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d^2 - 1
4578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a^20*d^5 +
a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a
^16*b^4*d^5 + 8*a^18*b^2*d^5) + (((tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*d^2 +
14272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2 + 126
16*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10
*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((192*a^2*b^24*d^4 + 1728*a^4*
b^22*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a^14*b^
12*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*d^4)/(
a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*
d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (tan(c + d*x)^(1/2)*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*(5
12*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a
^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24
*b^5*d^4 - 512*a^26*b^3*d^4))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)*(a^20*d^4 + a^4*b^16*d^4 + 8
*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a
^18*b^2*d^4)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2
*d)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*(-a
^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*(-a^5*b^3)^(
1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*1i)/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))/((9*b^14 + 60*a^2*
b^12 + 318*a^4*b^10 + 748*a^6*b^8 + 1505*a^8*b^6)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5
+ 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (((tan(c + d*x)^(
1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(a^20*
d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 +
28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^13*d
^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a^20
*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5
+ 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - (((tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^18*
d^2 + 14272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*d^2
+ 12616*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^1
0*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) + (((192*a^2*b^24*d^4 + 172
8*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 107520*a
^14*b^12*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b^2*
d^4)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^1
4*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - (tan(c + d*x)^(1/2)*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b
^2)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 2
1504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 460
8*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)*(a^20*d^4 + a^4*b^16*d
^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4
+ 8*a^18*b^2*d^4)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a
^9*b^2*d)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)
))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*(-a^5*
b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)) - (((tan(c + d*x
)^(1/2)*(18*a^2*b^15 - 9*b^17 - 71*a^4*b^13 + 892*a^6*b^11 + 857*a^8*b^9 + 6802*a^10*b^7 - 1257*a^12*b^5))/(a^
20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^
4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) + (((90*a*b^19*d^2 + 846*a^3*b^17*d^2 + 1714*a^5*b^15*d^2 + 3606*a^7*b^1
3*d^2 - 14578*a^9*b^11*d^2 - 34486*a^11*b^9*d^2 - 14970*a^13*b^7*d^2 + 2258*a^15*b^5*d^2 - 32*a^17*b^3*d^2)/(a
^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d
^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (((tan(c + d*x)^(1/2)*(72*a*b^22*d^2 + 576*a^3*b^20*d^2 + 5024*a^5*b^
18*d^2 + 14272*a^7*b^16*d^2 + 27824*a^9*b^14*d^2 + 53184*a^11*b^12*d^2 + 70240*a^13*b^10*d^2 + 47680*a^15*b^8*
d^2 + 12616*a^17*b^6*d^2 - 64*a^21*b^2*d^2))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*
a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((192*a^2*b^24*d^4 +
1728*a^4*b^22*d^4 + 8320*a^6*b^20*d^4 + 27264*a^8*b^18*d^4 + 62592*a^10*b^16*d^4 + 99456*a^12*b^14*d^4 + 10752
0*a^14*b^12*d^4 + 76800*a^16*b^10*d^4 + 33984*a^18*b^8*d^4 + 7872*a^20*b^6*d^4 + 384*a^22*b^4*d^4 - 128*a^24*b
^2*d^4)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*
a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (tan(c + d*x)^(1/2)*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^
2*b^2)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4
+ 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 -
4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)*(a^20*d^4 + a^4*b^1
6*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*
d^4 + 8*a^18*b^2*d^4)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d +
3*a^9*b^2*d)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2
*d)))*(-a^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d)))*(-a
^5*b^3)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d))))*(-a^5*b^3)^
(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*1i)/(4*(a^11*d + a^5*b^6*d + 3*a^7*b^4*d + 3*a^9*b^2*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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