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]

((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)

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Rubi [A]  time = 0.982629, antiderivative size = 268, normalized size of antiderivative = 1., 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 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 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]

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 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 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 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 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]

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 = 5.40464, 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))

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Maple [B]  time = 0.415, size = 2818, normalized size = 10.5 \begin{align*} \text{output too large to display} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 3.22973, size = 2570, normalized size = 9.59 \begin{align*} \text{result too large to display} \end{align*}

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*(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))*e^(I*d
*x + I*c) + 12*sqrt(1/2)*(a^2*d*e^(8*I*d*x + 8*I*c) - 2*a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))
*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log((2*I*sqrt(1/2)*a^2*d*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*e^(2*I*d
*x + 2*I*c) + sqrt(2)*((I*A + B)*e^(2*I*d*x + 2*I*c) + I*A + B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I
*c))*e^(-I*d*x - I*c)/(4*I*A + 4*B)) - 12*sqrt(1/2)*(a^2*d*e^(8*I*d*x + 8*I*c) - 2*a^2*d*e^(6*I*d*x + 6*I*c) +
a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((A^2 - 2*I*A*B - B^2)/(a^3*d^2))*log((-2*I*sqrt(1/2)*a^2*d*sqrt((A^2 - 2*I*A*
B - B^2)/(a^3*d^2))*e^(2*I*d*x + 2*I*c) + sqrt(2)*((I*A + B)*e^(2*I*d*x + 2*I*c) + I*A + B)*sqrt(a/(e^(2*I*d*x
+ 2*I*c) + 1))*e^(I*d*x + I*c))*e^(-I*d*x - I*c)/(4*I*A + 4*B)) - 3*(a^2*d*e^(8*I*d*x + 8*I*c) - 2*a^2*d*e^(6
*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2))*log(1/4*(4*sqrt(2
)*((4048*I*A - 2112*B)*e^(2*I*d*x + 2*I*c) + 4048*I*A - 2112*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I
*c) + (1056*I*a^2*d*e^(2*I*d*x + 2*I*c) + 352*I*a^2*d)*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2)))/((-116
61*I*A + 6084*B)*e^(2*I*d*x + 2*I*c) + 11661*I*A - 6084*B)) + 3*(a^2*d*e^(8*I*d*x + 8*I*c) - 2*a^2*d*e^(6*I*d*
x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2))*log(1/4*(4*sqrt(2)*((4
048*I*A - 2112*B)*e^(2*I*d*x + 2*I*c) + 4048*I*A - 2112*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c) +
(-1056*I*a^2*d*e^(2*I*d*x + 2*I*c) - 352*I*a^2*d)*sqrt((529*A^2 + 552*I*A*B - 144*B^2)/(a^3*d^2)))/((-11661*I
*A + 6084*B)*e^(2*I*d*x + 2*I*c) + 11661*I*A - 6084*B)))/(a^2*d*e^(8*I*d*x + 8*I*c) - 2*a^2*d*e^(6*I*d*x + 6*I
*c) + a^2*d*e^(4*I*d*x + 4*I*c))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

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]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)