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

Optimal. Leaf size=358 \[ \frac {\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 a^2+3 b^2}{a^2 d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d \left (a^2+b^2\right )^2} \]

[Out]

-b^(5/2)*(7*a^2+3*b^2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(5/2)/(a^2+b^2)^2/d-1/2*(a^2+2*a*b-b^2)*arct
an(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)-1/2*(a^2+2*a*b-b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(
a^2+b^2)^2/d*2^(1/2)-1/4*(a^2-2*a*b-b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+1/4*(
a^2-2*a*b-b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+(-2*a^2-3*b^2)/a^2/(a^2+b^2)/d/
tan(d*x+c)^(1/2)+b^2/a/(a^2+b^2)/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.76, antiderivative size = 358, 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} \[ -\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 a^2+3 b^2}{a^2 d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*
ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*S
qrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x
]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Ta
n[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2 + 3*b^2)/(a^2*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]) + b^2/(a*(a^
2 + b^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))^2} \, dx &=\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (2 a^2+3 b^2\right )-a b \tan (c+d x)+\frac {3}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \int \frac {\frac {1}{4} b \left (4 a^2+3 b^2\right )+\frac {1}{2} a^3 \tan (c+d x)+\frac {1}{4} b \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )}\\ &=-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \int \frac {a^3 b+\frac {1}{2} a^2 \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )^2}-\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \int \frac {1+\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 {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {4 \operatorname {Subst}\left (\int \frac {a^3 b+\frac {1}{2} a^2 \left (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 )^2 d}-\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 a^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\left (a^2-2 a b-b^2\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 )^2 d}-\frac {\left (a^2+2 a b-b^2\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 )^2 d}-\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (a^2-2 a b-b^2\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 )^2 d}-\frac {\left (a^2-2 a b-b^2\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 )^2 d}-\frac {\left (a^2+2 a b-b^2\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 )^2 d}-\frac {\left (a^2+2 a b-b^2\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 )^2 d}\\ &=-\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 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 )^2 d}+\frac {\left (a^2-2 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 )^2 d}-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (a^2+2 a b-b^2\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 )^2 d}+\frac {\left (a^2+2 a b-b^2\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 )^2 d}\\ &=\frac {\left (a^2+2 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 )^2 d}-\frac {\left (a^2+2 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 )^2 d}-\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 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 )^2 d}+\frac {\left (a^2-2 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 )^2 d}-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 1.79, size = 195, normalized size = 0.54 \[ -\frac {\frac {2 a^2+3 b^2}{a \sqrt {\tan (c+d x)}}+\frac {(-1)^{3/4} a \left ((a+i b)^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-(a-i b)^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )}-\frac {b^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)) + ((-1)^(3/4)*
a*((a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a
^2 + b^2) + (2*a^2 + 3*b^2)/(a*Sqrt[Tan[c + d*x]]) - b^2/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))/(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)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

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

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))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.23, size = 576, normalized size = 1.61 \[ -\frac {b^{3} \left (\sqrt {\tan }\left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{5} \left (\sqrt {\tan }\left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {7 b^{3} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right )}{d \left (a^{2}+b^{2}\right )^{2} \sqrt {a b}}-\frac {3 b^{5} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right )}{d \left (a^{2}+b^{2}\right )^{2} a^{2} \sqrt {a b}}-\frac {2}{d \,a^{2} \sqrt {\tan \left (d x +c \right )}}-\frac {a b \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {\sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) a^{2}}{4 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b^{2}}{4 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}} \]

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))^2,x)

[Out]

-1/d*b^3/(a^2+b^2)^2*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))-1/d*b^5/(a^2+b^2)^2/a^2*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c
))-7/d*b^3/(a^2+b^2)^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))-3/d*b^5/(a^2+b^2)^2/a^2/(a*b)^(1/2)*
arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))-2/d/a^2/tan(d*x+c)^(1/2)-1/d/(a^2+b^2)^2*a*b*2^(1/2)*arctan(-1+2^(1/2)*
tan(d*x+c)^(1/2))-1/2/d/(a^2+b^2)^2*a*b*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)))-1/d/(a^2+b^2)^2*a*b*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))-1/4/d/(a^2+b^2)^2*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+1/4/d/(a^2+b^2)^2*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^2-1/2/d/(a^2+b^2)^2
*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2+1/2/d/(a^2+b^2)^2*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))
*b^2-1/2/d/(a^2+b^2)^2*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2+1/2/d/(a^2+b^2)^2*2^(1/2)*arctan(1+2^(1/
2)*tan(d*x+c)^(1/2))*b^2

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maxima [A]  time = 0.57, size = 315, normalized size = 0.88 \[ -\frac {\frac {4 \, {\left (7 \, a^{2} b^{3} + 3 \, b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (2 \, a^{3} + 2 \, a b^{2} + {\left (2 \, a^{2} b + 3 \, b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{5} + a^{3} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(7*a^2*b^3 + 3*b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^6 + 2*a^4*b^2 + a^2*b^4)*sqrt(a*b)) + (
2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 + 2*a*b -
b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*(a^2 - 2*a*b - b^2)*log(sqrt(2)*sqrt(tan(
d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a^2 - 2*a*b - b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1
))/(a^4 + 2*a^2*b^2 + b^4) + 4*(2*a^3 + 2*a*b^2 + (2*a^2*b + 3*b^3)*tan(d*x + c))/((a^4*b + a^2*b^3)*tan(d*x +
c)^(3/2) + (a^5 + a^3*b^2)*sqrt(tan(d*x + c))))/d

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mupad [B]  time = 7.76, size = 12063, normalized size = 33.70 \[ \text {result too large to display} \]

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))^2),x)

[Out]

atan(((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(tan(c + d*x)^(1/2)*(14
4*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^2
4*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) + (1i/(4*(a^4*d^2 +
b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(26496*a^25*b^14*d^6 - 1152*a^15*b^24*d^6 - 844
8*a^17*b^22*d^6 - 23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a^23*b^16*d^6 - (1i/(4*(a^4*d^2 + b^4*d^2 +
a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i
- 6*a^2*b^2*d^2)))^(1/2)*(768*a^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*d^8 + 133120*a^22*b^21*d^8
+ 261120*a^24*b^19*d^8 + 347136*a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11
*d^8 - 7680*a^34*b^9*d^8 - 12032*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8 - tan(c + d*x)^(1/2)*(1i/
(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^
25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^
13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9)
) + tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7
+ 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*
d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7)) + 33984*a^27*b^12*d^6 + 1862
4*a^29*b^10*d^6 + 5376*a^31*b^8*d^6 + 1152*a^33*b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6))*1i + (1i/(4*(a^
4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(tan(c + d*x)^(1/2)*(144*a^14*b^23*d^5
+ 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24*b^13*d^5 - 563
2*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*
d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(26496*a^25*b^14*d^6 - 1152*a^15*b^24*d^6 - 8448*a^17*b^22*d^6
- 23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a^23*b^16*d^6 - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i -
a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2
)))^(1/2)*(768*a^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b
^19*d^8 + 347136*a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34
*b^9*d^8 - 12032*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8 + tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^
4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a
^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a
^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9)) - tan(c + d*x)
^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*b^
18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 3200*a^33*
b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7)) + 33984*a^27*b^12*d^6 + 18624*a^29*b^10*d^6
+ 5376*a^31*b^8*d^6 + 1152*a^33*b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6))*1i)/((1i/(4*(a^4*d^2 + b^4*d^2
+ a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(tan(c + d*x)^(1/2)*(144*a^14*b^23*d^5 + 1248*a^16*b^21
*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24*b^13*d^5 - 5632*a^26*b^11*d^5
- 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d
^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(26496*a^25*b^14*d^6 - 1152*a^15*b^24*d^6 - 8448*a^17*b^22*d^6 - 23776*a^19*b^2
0*d^6 - 29664*a^21*b^18*d^6 - 6528*a^23*b^16*d^6 - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6
*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(768*a
^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 347136
*a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34*b^9*d^8 - 12032
*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8 - tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^
2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 5
6320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 5
6320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9)) + tan(c + d*x)^(1/2)*(1152*a^1
5*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*
a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^
35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7)) + 33984*a^27*b^12*d^6 + 18624*a^29*b^10*d^6 + 5376*a^31*b^8*
d^6 + 1152*a^33*b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6)) - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^
3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(tan(c + d*x)^(1/2)*(144*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^
19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5
- 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d
^2)))^(1/2)*(26496*a^25*b^14*d^6 - 1152*a^15*b^24*d^6 - 8448*a^17*b^22*d^6 - 23776*a^19*b^20*d^6 - 29664*a^21*
b^18*d^6 - 6528*a^23*b^16*d^6 - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/
2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(768*a^16*b^27*d^8 + 8704
*a^18*b^25*d^8 + 44288*a^20*b^23*d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 347136*a^26*b^17*d^8 + 31
1808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34*b^9*d^8 - 12032*a^36*b^7*d^8 - 409
6*a^38*b^5*d^8 - 512*a^40*b^3*d^8 + tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i
- 6*a^2*b^2*d^2)))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9
+ 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 -
22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9)) - tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*
a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337
792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^
37*b^4*d^7 + 64*a^39*b^2*d^7)) + 33984*a^27*b^12*d^6 + 18624*a^29*b^10*d^6 + 5376*a^31*b^8*d^6 + 1152*a^33*b^6
*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6)) + 144*a^14*b^21*d^4 + 1296*a^16*b^19*d^4 + 4880*a^18*b^17*d^4 + 10
000*a^20*b^15*d^4 + 12080*a^22*b^13*d^4 + 8624*a^24*b^11*d^4 + 3376*a^26*b^9*d^4 + 560*a^28*b^7*d^4))*(1i/(4*(
a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*2i + atan((((tan(c + d*x)^(1/2)*(144*
a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24*
b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5))/2 + ((1/(a^4*d^2*1i +
b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(13248*a^25*b^14*d^6 - 576*a^15*b^24*d^6 - 42
24*a^17*b^22*d^6 - 11888*a^19*b^20*d^6 - 14832*a^21*b^18*d^6 - 3264*a^23*b^16*d^6 - ((1/(a^4*d^2*1i + b^4*d^2*
1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(((1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*
d^2 - a^2*b^2*d^2*6i))^(1/2)*(384*a^16*b^27*d^8 - (tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^
2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320
*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320
*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 4352*a^18*b^25*d^8 + 22144*a^2
0*b^23*d^8 + 66560*a^22*b^21*d^8 + 130560*a^24*b^19*d^8 + 173568*a^26*b^17*d^8 + 155904*a^28*b^15*d^8 + 89088*
a^30*b^13*d^8 + 24960*a^32*b^11*d^8 - 3840*a^34*b^9*d^8 - 6016*a^36*b^7*d^8 - 2048*a^38*b^5*d^8 - 256*a^40*b^3
*d^8))/2 + (tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b
^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^
31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7))/2))/2 + 16992*a^27*b
^12*d^6 + 9312*a^29*b^10*d^6 + 2688*a^31*b^8*d^6 + 576*a^33*b^6*d^6 + 144*a^35*b^4*d^6 + 16*a^37*b^2*d^6))/2)*
(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*1i + ((tan(c + d*x)^(1/2)*(14
4*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^2
4*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5))/2 - ((1/(a^4*d^2*1i
+ b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(13248*a^25*b^14*d^6 - 576*a^15*b^24*d^6 -
4224*a^17*b^22*d^6 - 11888*a^19*b^20*d^6 - 14832*a^21*b^18*d^6 - 3264*a^23*b^16*d^6 - ((1/(a^4*d^2*1i + b^4*d^
2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(((1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*
b*d^2 - a^2*b^2*d^2*6i))^(1/2)*((tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 -
a^2*b^2*d^2*6i))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 8
4480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22
528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 384*a^16*b^27*d^8 + 4352*a^18*b^25*d^8 + 22144*a
^20*b^23*d^8 + 66560*a^22*b^21*d^8 + 130560*a^24*b^19*d^8 + 173568*a^26*b^17*d^8 + 155904*a^28*b^15*d^8 + 8908
8*a^30*b^13*d^8 + 24960*a^32*b^11*d^8 - 3840*a^34*b^9*d^8 - 6016*a^36*b^7*d^8 - 2048*a^38*b^5*d^8 - 256*a^40*b
^3*d^8))/2 - (tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21
*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*
a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7))/2))/2 + 16992*a^27
*b^12*d^6 + 9312*a^29*b^10*d^6 + 2688*a^31*b^8*d^6 + 576*a^33*b^6*d^6 + 144*a^35*b^4*d^6 + 16*a^37*b^2*d^6))/2
)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*1i)/(((tan(c + d*x)^(1/2)*(
144*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a
^24*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5))/2 + ((1/(a^4*d^2*
1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(13248*a^25*b^14*d^6 - 576*a^15*b^24*d^6
- 4224*a^17*b^22*d^6 - 11888*a^19*b^20*d^6 - 14832*a^21*b^18*d^6 - 3264*a^23*b^16*d^6 - ((1/(a^4*d^2*1i + b^4*
d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(((1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^
3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(384*a^16*b^27*d^8 - (tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^
3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 5
6320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 5
6320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 4352*a^18*b^25*d^8 + 22144
*a^20*b^23*d^8 + 66560*a^22*b^21*d^8 + 130560*a^24*b^19*d^8 + 173568*a^26*b^17*d^8 + 155904*a^28*b^15*d^8 + 89
088*a^30*b^13*d^8 + 24960*a^32*b^11*d^8 - 3840*a^34*b^9*d^8 - 6016*a^36*b^7*d^8 - 2048*a^38*b^5*d^8 - 256*a^40
*b^3*d^8))/2 + (tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^
21*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 3763
2*a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7))/2))/2 + 16992*a^
27*b^12*d^6 + 9312*a^29*b^10*d^6 + 2688*a^31*b^8*d^6 + 576*a^33*b^6*d^6 + 144*a^35*b^4*d^6 + 16*a^37*b^2*d^6))
/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2) - ((tan(c + d*x)^(1/2)*(1
44*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^
24*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5))/2 - ((1/(a^4*d^2*1
i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(13248*a^25*b^14*d^6 - 576*a^15*b^24*d^6 -
4224*a^17*b^22*d^6 - 11888*a^19*b^20*d^6 - 14832*a^21*b^18*d^6 - 3264*a^23*b^16*d^6 - ((1/(a^4*d^2*1i + b^4*d
^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(((1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3
*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*((tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 -
a^2*b^2*d^2*6i))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 +
84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 2
2528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 384*a^16*b^27*d^8 + 4352*a^18*b^25*d^8 + 22144*
a^20*b^23*d^8 + 66560*a^22*b^21*d^8 + 130560*a^24*b^19*d^8 + 173568*a^26*b^17*d^8 + 155904*a^28*b^15*d^8 + 890
88*a^30*b^13*d^8 + 24960*a^32*b^11*d^8 - 3840*a^34*b^9*d^8 - 6016*a^36*b^7*d^8 - 2048*a^38*b^5*d^8 - 256*a^40*
b^3*d^8))/2 - (tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^2
1*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632
*a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7))/2))/2 + 16992*a^2
7*b^12*d^6 + 9312*a^29*b^10*d^6 + 2688*a^31*b^8*d^6 + 576*a^33*b^6*d^6 + 144*a^35*b^4*d^6 + 16*a^37*b^2*d^6))/
2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2) + 144*a^14*b^21*d^4 + 1296
*a^16*b^19*d^4 + 4880*a^18*b^17*d^4 + 10000*a^20*b^15*d^4 + 12080*a^22*b^13*d^4 + 8624*a^24*b^11*d^4 + 3376*a^
26*b^9*d^4 + 560*a^28*b^7*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/
2)*1i - (2/a + (tan(c + d*x)*(2*a^2*b + 3*b^3))/(a^2*(a^2 + b^2)))/(a*d*tan(c + d*x)^(1/2) + b*d*tan(c + d*x)^
(3/2)) + (atan((((7*a^2 + 3*b^2)*(tan(c + d*x)^(1/2)*(144*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*
d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 -
560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) + ((7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(26496*a^25*b^14*d^6 - 8448*a^17*b^22*
d^6 - 23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a^23*b^16*d^6 - 1152*a^15*b^24*d^6 + 33984*a^27*b^12*d^
6 + 18624*a^29*b^10*d^6 + 5376*a^31*b^8*d^6 + 1152*a^33*b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6 - ((tan(c
+ d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*
a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 320
0*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7) + ((7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(7
68*a^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 34
7136*a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34*b^9*d^8 - 1
2032*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8 - (tan(c + d*x)^(1/2)*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2
)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 +
67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 51
20*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2
*d)))*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*
b^2*d)))*(-a^5*b^5)^(1/2)*1i)/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d)) + ((7*a^2 + 3*b^2)*(tan(c + d*x)^(1/2)*(14
4*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^2
4*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) - ((7*a^2 + 3*b^2)*(
-a^5*b^5)^(1/2)*(26496*a^25*b^14*d^6 - 8448*a^17*b^22*d^6 - 23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a
^23*b^16*d^6 - 1152*a^15*b^24*d^6 + 33984*a^27*b^12*d^6 + 18624*a^29*b^10*d^6 + 5376*a^31*b^8*d^6 + 1152*a^33*
b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6 + ((tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7
+ 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*
d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 6
4*a^39*b^2*d^7) - ((7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(768*a^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*
d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 347136*a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*
b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34*b^9*d^8 - 12032*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8
+ (tan(c + d*x)^(1/2)*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b
^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b
^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(2*(a^9*d + a^5*b^4
*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d)))*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2))/(2*(a^9*d + a^5*
b^4*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d)))*(-a^5*b^5)^(1/2)*1i)/(2*(a^9*d + a^5*b^4*d + 2*
a^7*b^2*d)))/(144*a^14*b^21*d^4 + 1296*a^16*b^19*d^4 + 4880*a^18*b^17*d^4 + 10000*a^20*b^15*d^4 + 12080*a^22*b
^13*d^4 + 8624*a^24*b^11*d^4 + 3376*a^26*b^9*d^4 + 560*a^28*b^7*d^4 + ((7*a^2 + 3*b^2)*(tan(c + d*x)^(1/2)*(14
4*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^2
4*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32*a^32*b^5*d^5) + ((7*a^2 + 3*b^2)*(
-a^5*b^5)^(1/2)*(26496*a^25*b^14*d^6 - 8448*a^17*b^22*d^6 - 23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a
^23*b^16*d^6 - 1152*a^15*b^24*d^6 + 33984*a^27*b^12*d^6 + 18624*a^29*b^10*d^6 + 5376*a^31*b^8*d^6 + 1152*a^33*
b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6 - ((tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7
+ 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*
d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 3200*a^33*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 6
4*a^39*b^2*d^7) + ((7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(768*a^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*
d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 347136*a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*
b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34*b^9*d^8 - 12032*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8
- (tan(c + d*x)^(1/2)*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b
^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b
^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(2*(a^9*d + a^5*b^4
*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d)))*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2))/(2*(a^9*d + a^5*
b^4*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d)))*(-a^5*b^5)^(1/2))/(2*(a^9*d + a^5*b^4*d + 2*a^7
*b^2*d)) - ((7*a^2 + 3*b^2)*(tan(c + d*x)^(1/2)*(144*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 +
6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24*b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a
^30*b^7*d^5 + 32*a^32*b^5*d^5) - ((7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(26496*a^25*b^14*d^6 - 8448*a^17*b^22*d^6 -
23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a^23*b^16*d^6 - 1152*a^15*b^24*d^6 + 33984*a^27*b^12*d^6 + 1
8624*a^29*b^10*d^6 + 5376*a^31*b^8*d^6 + 1152*a^33*b^6*d^6 + 288*a^35*b^4*d^6 + 32*a^37*b^2*d^6 + ((tan(c + d*
x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*
b^18*d^7 + 443136*a^25*b^16*d^7 + 337792*a^27*b^14*d^7 + 156160*a^29*b^12*d^7 + 37632*a^31*b^10*d^7 + 3200*a^3
3*b^8*d^7 + 704*a^35*b^6*d^7 + 512*a^37*b^4*d^7 + 64*a^39*b^2*d^7) - ((7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(768*a^
16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b^23*d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 347136*
a^26*b^17*d^8 + 311808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11*d^8 - 7680*a^34*b^9*d^8 - 12032*
a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d^8 + (tan(c + d*x)^(1/2)*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*(51
2*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584
*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^
40*b^5*d^9 - 512*a^42*b^3*d^9))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d)))
*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d))))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d
)))*(-a^5*b^5)^(1/2))/(2*(a^9*d + a^5*b^4*d + 2*a^7*b^2*d))))*(7*a^2 + 3*b^2)*(-a^5*b^5)^(1/2)*1i)/(a^9*d + a^
5*b^4*d + 2*a^7*b^2*d)

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

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))**2,x)

[Out]

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

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