\(\int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) [507]

Optimal result

Integrand size = 34, antiderivative size = 105 \[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \sqrt [4]{-1} a (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 a (i A+B)}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a (A-i B)}{d \sqrt {\cot (c+d x)}} \]

[Out]

-2*(-1)^(1/4)*a*(I*A+B)*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d+2/5*I*a*B/d/cot(d*x+c)^(5/2)+2/3*a*(I*A+B)/d/co
t(d*x+c)^(3/2)+2*a*(A-I*B)/d/cot(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3662, 3672, 3610, 3614, 214} \[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \sqrt [4]{-1} a (B+i A) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a (B+i A)}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a (A-i B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)} \]

[In]

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

[Out]

(-2*(-1)^(1/4)*a*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (((2*I)/5)*a*B)/(d*Cot[c + d*x]^(5/2))
+ (2*a*(I*A + B))/(3*d*Cot[c + d*x]^(3/2)) + (2*a*(A - I*B))/(d*Sqrt[Cot[c + d*x]])

Rule 214

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

Rule 3610

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

Rule 3614

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2*(c^2/f), S
ubst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]

Rule 3662

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d
+ c*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ
[n]

Rule 3672

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x)) (B+A \cot (c+d x))}{\cot ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\int \frac {a (i A+B)+a (A-i B) \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 a (i A+B)}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\int \frac {a (A-i B)-a (i A+B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 a (i A+B)}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a (A-i B)}{d \sqrt {\cot (c+d x)}}+\int \frac {-a (i A+B)-a (A-i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 a (i A+B)}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a (A-i B)}{d \sqrt {\cot (c+d x)}}+\frac {\left (2 a^2 (i A+B)^2\right ) \text {Subst}\left (\int \frac {1}{a (i A+B)-a (A-i B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = -\frac {2 \sqrt [4]{-1} a (i A+B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 i a B}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 a (i A+B)}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a (A-i B)}{d \sqrt {\cot (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.99 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 i a \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (3 B \tan ^{\frac {5}{2}}(c+d x)+5 (A-i B) \left (-3 (-1)^{3/4} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)} (-3 i+\tan (c+d x))\right )\right )}{15 d} \]

[In]

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

[Out]

(((2*I)/15)*a*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*B*Tan[c + d*x]^(5/2) + 5*(A - I*B)*(-3*(-1)^(3/4)*ArcTa
n[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*(-3*I + Tan[c + d*x]))))/d

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (85 ) = 170\).

Time = 0.40 (sec) , antiderivative size = 470, normalized size of antiderivative = 4.48

method	result	size
derivativedivides	\(-\frac {a \left (120 i B \sqrt {\tan \left (d x +c \right )}+30 i A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-40 i A \tan \left (d x +c \right )^{\frac {3}{2}}-30 i B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-30 i B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+15 i A \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right )-15 i B \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+30 i A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-40 B \tan \left (d x +c \right )^{\frac {3}{2}}+15 A \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+30 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-24 i B \tan \left (d x +c \right )^{\frac {5}{2}}+30 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+15 B \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right )-120 A \sqrt {\tan \left (d x +c \right )}\right )}{60 d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right )^{\frac {3}{2}}}\)	\(470\)
default	\(-\frac {a \left (120 i B \sqrt {\tan \left (d x +c \right )}+30 i A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-40 i A \tan \left (d x +c \right )^{\frac {3}{2}}-30 i B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-30 i B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+15 i A \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right )-15 i B \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+30 i A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-40 B \tan \left (d x +c \right )^{\frac {3}{2}}+15 A \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+30 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-24 i B \tan \left (d x +c \right )^{\frac {5}{2}}+30 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+30 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+15 B \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right )-120 A \sqrt {\tan \left (d x +c \right )}\right )}{60 d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right )^{\frac {3}{2}}}\)	\(470\)

[In]

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

[Out]

-1/60*a/d*(120*I*B*tan(d*x+c)^(1/2)+30*I*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)-40*I*A*tan(d*x+c)^(3/2)-
30*I*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)-30*I*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)+15*I*A*ln
(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)-15*I*B*ln(-(1+2^(1/
2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)+30*I*A*arctan(-1+2^(1/2)*tan(
d*x+c)^(1/2))*2^(1/2)-40*B*tan(d*x+c)^(3/2)+15*A*2^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*
tan(d*x+c)^(1/2)-tan(d*x+c)-1))+30*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+30*A*2^(1/2)*arctan(-1+2^(1/2)
*tan(d*x+c)^(1/2))-24*I*B*tan(d*x+c)^(5/2)+30*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+30*B*2^(1/2)*arctan
(-1+2^(1/2)*tan(d*x+c)^(1/2))+15*B*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(
1/2)+tan(d*x+c)))-120*A*tan(d*x+c)^(1/2))/(1/tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (81) = 162\).

Time = 0.27 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.59 \[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {15 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{2}}{d^{2}}} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) - 15 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{2}}{d^{2}}} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (i \, A + B\right )} a}\right ) + 4 \, {\left ({\left (20 i \, A + 23 \, B\right )} a e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (10 i \, A + B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-20 i \, A - 11 \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-10 i \, A - 13 \, B\right )} a\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{30 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-1/30*(15*(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-(I*A^2 + 2*A*B
- I*B^2)*a^2/d^2)*log(-2*((A - I*B)*a*e^(2*I*d*x + 2*I*c) + (d*e^(2*I*d*x + 2*I*c) - d)*sqrt(-(I*A^2 + 2*A*B
- I*B^2)*a^2/d^2)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/((I*A + B)
*a)) - 15*(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-(I*A^2 + 2*A*B
- I*B^2)*a^2/d^2)*log(-2*((A - I*B)*a*e^(2*I*d*x + 2*I*c) - (d*e^(2*I*d*x + 2*I*c) - d)*sqrt(-(I*A^2 + 2*A*B
- I*B^2)*a^2/d^2)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/((I*A + B)
*a)) + 4*((20*I*A + 23*B)*a*e^(6*I*d*x + 6*I*c) + (10*I*A + B)*a*e^(4*I*d*x + 4*I*c) + (-20*I*A - 11*B)*a*e^(2
*I*d*x + 2*I*c) + (-10*I*A - 13*B)*a)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(d*e^(6*I*d
*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (81) = 162\).

Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.82 \[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {8 \, {\left (-3 i \, B a - \frac {5 \, {\left (i \, A + B\right )} a}{\tan \left (d x + c\right )} - \frac {15 \, {\left (A - i \, B\right )} a}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 15 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a}{60 \, d} \]

[In]

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

[Out]

-1/60*(8*(-3*I*B*a - 5*(I*A + B)*a/tan(d*x + c) - 15*(A - I*B)*a/tan(d*x + c)^2)*tan(d*x + c)^(5/2) + 15*(2*sq
rt(2)*(-(I + 1)*A + (I - 1)*B)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(-(I + 1)*A +
(I - 1)*B)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt(2)*((I - 1)*A + (I + 1)*B)*log(sqrt(2)
/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*((I - 1)*A + (I + 1)*B)*log(-sqrt(2)/sqrt(tan(d*x + c)) +
1/tan(d*x + c) + 1))*a)/d

Giac [F]

\[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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