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

Optimal. Leaf size=217 \[ \frac{(-2 B+3 i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

[Out]

(((3*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x])/(3*d*(a + I*a*Tan[c + d*x])^(
3/2)) + ((13*A + (7*I)*B)*Cot[c + d*x])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((7*A + (3*I)*B)*Cot[c + d*x]*Sqr
t[a + I*a*Tan[c + d*x]])/(2*a^2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.80406, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{(-2 B+3 i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Ta
n[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x])/(3*d*(a + I*a*Tan[c + d*x])^(
3/2)) + ((13*A + (7*I)*B)*Cot[c + d*x])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((7*A + (3*I)*B)*Cot[c + d*x]*Sqr
t[a + I*a*Tan[c + d*x]])/(2*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 ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{\int \frac{\cot ^2(c+d x) \left (a (4 A+i B)-\frac{5}{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 (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\int \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \left (\frac{3}{2} a^2 (7 A+3 i B)-\frac{3}{4} a^2 (13 i A-7 B) \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{\int \cot (c+d x) \sqrt{a+i a \tan (c+d x)} \left (-\frac{3}{2} a^3 (3 i A-2 B)-\frac{3}{4} a^3 (7 A+3 i B) \tan (c+d x)\right ) \, dx}{3 a^5}\\ &=\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}-\frac{(3 i A-2 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)} \, dx}{2 a^3}-\frac{(A-i B) \int \sqrt{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac{(A+i B) \cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}-\frac{(3 i A-2 B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a d}+\frac{(i A+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{(i A+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 (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}-\frac{(3 A+2 i B) \operatorname{Subst}\left (\int \frac{1}{i-\frac{i x^2}{a}} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{a^2 d}\\ &=\frac{(3 i A-2 B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}+\frac{(i A+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 (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{(13 A+7 i B) \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}-\frac{(7 A+3 i B) \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}\\ \end{align*}

Mathematica [A]  time = 4.7885, size = 259, normalized size = 1.19 \[ \frac{\sqrt{\sec (c+d x)} (A+B \tan (c+d x)) \left (\sqrt{2} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2} \left (1+e^{2 i (c+d x)}\right )^{3/2} \left ((B+i A) \sinh ^{-1}\left (e^{i (c+d x)}\right )+2 \sqrt{2} (-2 B+3 i A) \tanh ^{-1}\left (\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right )\right )-\frac{\csc (c+d x) ((-11 B+29 i A) \sin (2 (c+d x))+9 (3 A+i B) \cos (2 (c+d x))-3 (5 A+3 i B))}{3 \sqrt{\sec (c+d x)}}\right )}{4 d (a+i a \tan (c+d x))^{3/2} (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^2*(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)
*((I*A + B)*ArcSinh[E^(I*(c + d*x))] + 2*Sqrt[2]*((3*I)*A - 2*B)*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^
((2*I)*(c + d*x))]]) - (Csc[c + d*x]*(-3*(5*A + (3*I)*B) + 9*(3*A + I*B)*Cos[2*(c + d*x)] + ((29*I)*A - 11*B)*
Sin[2*(c + d*x)]))/(3*Sqrt[Sec[c + d*x]]))*(A + B*Tan[c + d*x]))/(4*d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I
*a*Tan[c + d*x])^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.432, size = 2818, normalized size = 13. \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)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x)

[Out]

-1/24/d/a^2*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*(3*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*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)-16*B*cos(d*x+c)^6-84*A*cos(d*x+c)*sin(d*x+c)+16*I*A*cos(d*x+c)^6+36*I*A*cos(d*x+c)^4-52*I*A*cos(d*x+c
)^2+16*A*cos(d*x+c)^5*sin(d*x+c)+44*A*cos(d*x+c)^3*sin(d*x+c)-3*I*A*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))-18*I*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))+12*I*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)/si
n(d*x+c))+12*I*B*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)
)-18*I*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))+12*I
*B*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))+18*I*A*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))-12*I*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))+18*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)+cos(d*x+c)-1)/sin(d*x+c))*sin(d*x+c)-3*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arct
an(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)+3*B*cos(d*
x+c)^3*(-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)+3*B*cos(d*x+c)^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))*2^(1/2)-3*B*cos(d*x+c)*(-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)-3*A*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(I*cos(d*x+c)-I-s
in(d*x+c))/sin(d*x+c)/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)+18*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)-12*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))*sin(d*x+c)-18*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))-12*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))-18*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))+12*B*cos(d*x+c)^2*sin(d*x+c)*(-2*co
s(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))
+18*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))-12*I*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))+16*I
*B*cos(d*x+c)^5*sin(d*x+c)-36*I*B*cos(d*x+c)*sin(d*x+c)+20*I*B*cos(d*x+c)^3*sin(d*x+c)+18*A*cos(d*x+c)^3*(-2*c
os(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)
)+12*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))+18*A*c
os(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))+12*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))-18*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))-12*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))-18*I*A*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))-12*I*B*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+3*I*A*cos(d*x+c)^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))*2
^(1/2)+3*I*B*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))+3*I*A*cos(d*x+c)^3*(-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)-3*I*A
*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))-3*I*B*cos(d*x+c)^2*sin(d*x+c)*(-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)-12*cos
(d*x+c)^4*B+28*B*cos(d*x+c)^2)/(-1+cos(d*x+c)^2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.13325, size = 2230, normalized size = 10.28 \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)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/12*(sqrt(2)*((-28*I*A + 10*B)*e^(6*I*d*x + 6*I*c) + (-13*I*A + B)*e^(4*I*d*x + 4*I*c) + (16*I*A - 10*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) + 3*sqrt(1/2)*(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*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*sqrt(1/2)*(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*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^(6*I*d*x + 6*I*c) - a^2*d*e
^(4*I*d*x + 4*I*c))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2))*log((sqrt(2)*((528*I*A - 352*B)*e^(2*I*d*x + 2
*I*c) + 528*I*A - 352*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c) + 88*(3*a^2*d*e^(2*I*d*x + 2*I*c) +
a^2*d)*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2)))/((-1521*I*A + 1014*B)*e^(2*I*d*x + 2*I*c) + 1521*I*A - 10
14*B)) - 3*(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2))
*log((sqrt(2)*((528*I*A - 352*B)*e^(2*I*d*x + 2*I*c) + 528*I*A - 352*B)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I
*d*x + I*c) - 88*(3*a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(-(9*A^2 + 12*I*A*B - 4*B^2)/(a^3*d^2)))/((-1521*I*
A + 1014*B)*e^(2*I*d*x + 2*I*c) + 1521*I*A - 1014*B)))/(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))

________________________________________________________________________________________

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

[Out]

Exception raised: AttributeError

________________________________________________________________________________________

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 )^{2}}{{\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)^2*(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)^2/(I*a*tan(d*x + c) + a)^(3/2), x)