3.103 \(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=268 \[ \frac {(23 A+12 i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {7 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

[Out]

1/4*(23*A+12*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-1/4*(A-I*B)*arctanh(1/2*(a+I*a*tan(d*x+c
))^(1/2)*2^(1/2)/a^(1/2))/a^(3/2)/d*2^(1/2)+1/6*(17*A+11*I*B)*cot(d*x+c)^2/a/d/(a+I*a*tan(d*x+c))^(1/2)+7/4*(3
*I*A-2*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/a^2/d-1/6*(22*A+13*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/a^
2/d+1/3*(A+I*B)*cot(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.98, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3596, 3598, 3600, 3480, 206, 3599, 63, 208} \[ \frac {(23 A+12 i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {7 (-2 B+3 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

((23*A + (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*
a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x]^2)/(3*d*(a + I*a*Tan[c + d
*x])^(3/2)) + ((17*A + (11*I)*B)*Cot[c + d*x]^2)/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (7*((3*I)*A - 2*B)*Cot[c
+ d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^2*d) - ((22*A + (13*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/
(6*a^2*d)

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 206

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

Rule 208

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

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3596

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

Rule 3598

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

Rule 3599

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

Rule 3600

Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(A*b + a*B)/(b*c + a*d), Int[(a + b*Tan[e + f*x])^m, x], x] - Dist[(B*c
- A*d)/(b*c + a*d), Int[((a + b*Tan[e + f*x])^m*(a - b*Tan[e + f*x]))/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{
a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]

Rubi steps

\begin {align*} \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^3(c+d x) \left (a (5 A+2 i B)-\frac {7}{2} a (i A-B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (a^2 (22 A+13 i B)-\frac {5}{4} a^2 (17 i A-11 B) \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {21}{2} a^3 (3 i A-2 B)-\frac {3}{2} a^3 (22 A+13 i B) \tan (c+d x)\right ) \, dx}{6 a^5}\\ &=\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^4 (23 A+12 i B)+\frac {21}{4} a^4 (3 i A-2 B) \tan (c+d x)\right ) \, dx}{6 a^6}\\ &=\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}-\frac {(23 A+12 i B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{8 a^3}-\frac {(i A+B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}-\frac {(A-i B) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2 a d}-\frac {(23 A+12 i B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{8 a d}\\ &=-\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {(23 i A-12 B) \operatorname {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{4 a^2 d}\\ &=\frac {(23 A+12 i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A+i B) \cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(17 A+11 i B) \cot ^2(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 (3 i A-2 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 a^2 d}-\frac {(22 A+13 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.13, size = 283, normalized size = 1.06 \[ \frac {\sqrt {\sec (c+d x)} (A+B \tan (c+d x)) \left (\sqrt {2} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2} \left (1+e^{2 i (c+d x)}\right )^{3/2} \left (\sqrt {2} (23 A+12 i B) \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {1+e^{2 i (c+d x)}}}\right )-2 (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )+\frac {\csc ^2(c+d x) (-((50 A+29 i B) \cos (c+d x))+(38 A+29 i B) \cos (3 (c+d x))+6 \sin (c+d x) ((-9 B+12 i A) \cos (2 (c+d x))-9 i A+5 B))}{3 \sqrt {\sec (c+d x)}}\right )}{8 d (a+i a \tan (c+d x))^{3/2} (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(Sqrt[2]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)
*(-2*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(23*A + (12*I)*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 +
E^((2*I)*(c + d*x))]]) + (Csc[c + d*x]^2*(-((50*A + (29*I)*B)*Cos[c + d*x]) + (38*A + (29*I)*B)*Cos[3*(c + d*
x)] + 6*((-9*I)*A + 5*B + ((12*I)*A - 9*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(3*Sqrt[Sec[c + d*x]]))*(A + B*Tan
[c + d*x]))/(8*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2))

________________________________________________________________________________________

fricas [B]  time = 0.57, size = 901, normalized size = 3.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/48*(12*sqrt(1/2)*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt
((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log((sqrt(2)*sqrt(1/2)*(4*I*a^2*d*e^(2*I*d*x + 2*I*c) + 4*I*a^2*d)*sqrt(a/(e
^(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2)) + (4*I*A + 4*B)*a*e^(I*d*x + I*c))*e^(-I*d*x -
I*c)/(I*A + B)) - 12*sqrt(1/2)*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3
*I*c))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log((sqrt(2)*sqrt(1/2)*(-4*I*a^2*d*e^(2*I*d*x + 2*I*c) - 4*I*a^2*
d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2)) + (4*I*A + 4*B)*a*e^(I*d*x + I*c))*
e^(-I*d*x - I*c)/(I*A + B)) + 3*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x +
3*I*c))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2))*log(1/4*(4*(-1104*I*A + 576*B)*a^2*e^(2*I*d*x + 2*I*c)
+ 4*(-368*I*A + 192*B)*a^2 + sqrt(2)*(128*I*a^3*d*e^(3*I*d*x + 3*I*c) + 128*I*a^3*d*e^(I*d*x + I*c))*sqrt(a/(
e^(2*I*d*x + 2*I*c) + 1))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-23*I*A + 12*
B)) - 3*(a^2*d*e^(7*I*d*x + 7*I*c) - 2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))*sqrt((529*A^2 +
552*I*A*B - 144*B^2)/(a^3*d^2))*log(1/4*(4*(-1104*I*A + 576*B)*a^2*e^(2*I*d*x + 2*I*c) + 4*(-368*I*A + 192*B)*
a^2 + sqrt(2)*(-128*I*a^3*d*e^(3*I*d*x + 3*I*c) - 128*I*a^3*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1
))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2)))*e^(-2*I*d*x - 2*I*c)/(-23*I*A + 12*B)) + 4*sqrt(2)*((37*A
+ 28*I*B)*e^(8*I*d*x + 8*I*c) - 3*(11*A + 5*I*B)*e^(6*I*d*x + 6*I*c) - (50*A + 29*I*B)*e^(4*I*d*x + 4*I*c) + 3
*(7*A + 5*I*B)*e^(2*I*d*x + 2*I*c) + A + I*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(a^2*d*e^(7*I*d*x + 7*I*c) -
2*a^2*d*e^(5*I*d*x + 5*I*c) + a^2*d*e^(3*I*d*x + 3*I*c))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^3/(I*a*tan(d*x + c) + a)^(3/2), x)

________________________________________________________________________________________

maple [B]  time = 3.50, size = 2818, normalized size = 10.51 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x)

[Out]

1/48/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(-6*B*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+
c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(
1/2)+176*A*cos(d*x+c)^2-69*A*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*(
-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+32*I*A*cos(d*x+c)^5*sin(d*x+c)+136*I*A*cos(d*x+c)^3*sin(d*x+c)-
252*I*A*cos(d*x+c)*sin(d*x+c)-69*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))
^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-36*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c
)/(1+cos(d*x+c)))^(1/2))-120*A*cos(d*x+c)^4-88*B*cos(d*x+c)^3*sin(d*x+c)+6*A*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2
))*2^(1/2)+6*A*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*
x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*2^(1/2)+168*B*cos(d*x+c)*sin(d*x+c)-69*A*(-2*cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+36*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln
(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-32*B*cos(d*x+c)^5*sin(d*x+c)-32*
I*B*cos(d*x+c)^6-72*I*B*cos(d*x+c)^4+104*I*B*cos(d*x+c)^2+6*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c
)*2^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-69*I
*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))
+36*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2
)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+6*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*2^(1/2)*arctan(1/
2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-6*I*B*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*2^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*2^(1/2))-36*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*arctan(1/(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2))+69*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^3*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))
^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+36*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^3*arctan(1/
(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-69*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)*ln(-(-(-2*cos(d*x
+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-36*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos
(d*x+c)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+69*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*
arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+6*B*2^(1/2)*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arc
tan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+69*A*(-2*cos(d*x+
c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^2-36*B*(-2*cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos(d*x+c)^2-
36*B*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+c
os(d*x+c)-1)/sin(d*x+c))+36*B*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+
c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+69*A*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(
1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+36*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*arctan(1/(-2*
cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+69*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*ln(-(-(-2*cos(d*x+c
)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-6*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*
x+c)^3*2^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))
-69*A*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-6*A*2^(1/
2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)*2^(1/2))-6*I*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)*arctan(1/
2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-36*I*B*(-2*cos(d*x+c)/(
1+cos(d*x+c)))^(1/2)*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c)
)+6*I*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)*arctan(1/2*(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos
(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-6*A*cos(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*
(I*cos(d*x+c)-I-sin(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+69*A*(-2*cos(d*x+c)/(1+co
s(d*x+c)))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/s
in(d*x+c))+36*B*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*
x+c)))^(1/2))-32*A*cos(d*x+c)^6)/(cos(d*x+c)^2-1)/a^2

________________________________________________________________________________________

maxima [A]  time = 0.89, size = 259, normalized size = 0.97 \[ -\frac {a^{2} {\left (\frac {2 \, {\left (21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} {\left (3 \, A + 2 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (107 \, A + 68 i \, B\right )} a + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (17 \, A + 11 i \, B\right )} a^{2} + 4 \, {\left (A + i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5}} - \frac {3 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {7}{2}}} + \frac {3 \, {\left (23 \, A + 12 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

-1/24*a^2*(2*(21*(I*a*tan(d*x + c) + a)^3*(3*A + 2*I*B) - (I*a*tan(d*x + c) + a)^2*(107*A + 68*I*B)*a + 2*(I*a
*tan(d*x + c) + a)*(17*A + 11*I*B)*a^2 + 4*(A + I*B)*a^3)/((I*a*tan(d*x + c) + a)^(7/2)*a^3 - 2*(I*a*tan(d*x +
c) + a)^(5/2)*a^4 + (I*a*tan(d*x + c) + a)^(3/2)*a^5) - 3*sqrt(2)*(A - I*B)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*
tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a)))/a^(7/2) + 3*(23*A + 12*I*B)*log((sqrt(I*a*t
an(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(7/2))/d

________________________________________________________________________________________

mupad [B]  time = 8.41, size = 3106, normalized size = 11.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^(3/2),x)

[Out]

2*atanh((48*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4
*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*
B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2)*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^
2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/(B^3*a^2*d*3124i - 25296*A^3*a^2*d +
19048*A*B^2*a^2*d - A^2*B*a^2*d*38282i + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A
^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a + (B*d^3*((277729*A^4*a^6)/(4*d^4
) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52
i)/a) - (4216*A^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (50
41*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6)
- (73*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)
/a - (A^2*B*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4
- (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*
a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) + (113
6*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/
d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*B^2)/(
64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)/a - (A^2*B*
d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6
*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (
114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) - (A*B*d^2*(a + a*
tan(c + d*x)*1i)^(1/2)*((531*A^2)/(128*a^3*d^2) - ((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2
*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) - (73*B^2)/(64*a^3*d^2) + (A*
B*137i)/(32*a^3*d^2))^(1/2)*4448i)/((B^3*d*3124i)/a - (25296*A^3*d)/a + (19048*A*B^2*d)/a - (A^2*B*d*38282i)/a
+ (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^
4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*
B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4))*(-(2*d^2*((((531*A^2*a^3)/2
- 73*B^2*a^3)/d^2 + (A*B*a^3*274i)/d^2)^2 + 128*a^6*((((33*A*B^3)/8 + (253*A^3*B)/32)*1i)/d^4 - ((529*A^4)/64
+ (431*A^2*B^2)/64 + (9*B^4)/4)/d^4))^(1/2) - 531*A^2*a^3 + 146*B^2*a^3 - A*B*a^3*548i)/(128*a^6*d^2))^(1/2) +
2*atanh((48*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B
^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2) - (73
*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2)*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A
^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/(25296*A^3*a^2*d - B^3*a^2*d*3124i
- 19048*A*B^2*a^2*d + A^2*B*a^2*d*38282i + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*
A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a + (B*d^3*((277729*A^4*a^6)/(4*d^
4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*5
2i)/a) + (4216*A^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701
*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2)
- (73*B^2)/(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((25296*A^3*d)/a - (B^3*d*3124i)/a - (19048*A*B^2*d
)/a + (A^2*B*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4
- (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4
*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) - (11
36*B^2*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6
)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2) - (73*B^2)/
(64*a^3*d^2) + (A*B*137i)/(32*a^3*d^2))^(1/2))/((25296*A^3*d)/a - (B^3*d*3124i)/a - (19048*A*B^2*d)/a + (A^2*B
*d*38282i)/a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^
6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 -
(114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4) + (A*B*d^2*(a + a
*tan(c + d*x)*1i)^(1/2)*(((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^
6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)/(64*a^6) + (531*A^2)/(128*a^3*d^2) - (73*B^2)/(64*a^3*d^2) + (A
*B*137i)/(32*a^3*d^2))^(1/2)*4448i)/((25296*A^3*d)/a - (B^3*d*3124i)/a - (19048*A*B^2*d)/a + (A^2*B*d*38282i)/
a + (88*A*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d
^4 + (A^3*B*a^6*146506i)/d^4)^(1/2))/a^4 + (B*d^3*((277729*A^4*a^6)/(4*d^4) + (5041*B^4*a^6)/d^4 - (114701*A^2
*B^2*a^6)/d^4 - (A*B^3*a^6*39476i)/d^4 + (A^3*B*a^6*146506i)/d^4)^(1/2)*52i)/a^4))*((2*d^2*((((531*A^2*a^3)/2
- 73*B^2*a^3)/d^2 + (A*B*a^3*274i)/d^2)^2 + 128*a^6*((((33*A*B^3)/8 + (253*A^3*B)/32)*1i)/d^4 - ((529*A^4)/64
+ (431*A^2*B^2)/64 + (9*B^4)/4)/d^4))^(1/2) + 531*A^2*a^3 - 146*B^2*a^3 + A*B*a^3*548i)/(128*a^6*d^2))^(1/2) -
((A*a^2 + B*a^2*1i)/(3*d) - ((107*A + B*68i)*(a + a*tan(c + d*x)*1i)^2)/(12*d) + ((17*A*a + B*a*11i)*(a + a*t
an(c + d*x)*1i))/(6*d) + (7*(3*A + B*2i)*(a + a*tan(c + d*x)*1i)^3)/(4*a*d))/((a + a*tan(c + d*x)*1i)^(7/2) -
2*a*(a + a*tan(c + d*x)*1i)^(5/2) + a^2*(a + a*tan(c + d*x)*1i)^(3/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**(3/2),x)

[Out]

Integral((A + B*tan(c + d*x))*cot(c + d*x)**3/(I*a*(tan(c + d*x) - I))**(3/2), x)

________________________________________________________________________________________