\(\int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx\) [518]

Optimal result

Integrand size = 36, antiderivative size = 173 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {8 \sqrt [4]{-1} a^3 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {8 a^3 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)} \]

[Out]

8*(-1)^(1/4)*a^3*(A-I*B)*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-8/105*a^3*(21*A-23*I*B)/d/cot(d*x+c)^(3/2)+2/7
*I*a*B*(I*a+a*cot(d*x+c))^2/d/cot(d*x+c)^(7/2)-2/35*(7*A-11*I*B)*(I*a^3+a^3*cot(d*x+c))/d/cot(d*x+c)^(5/2)+8*a
^3*(I*A+B)/d/cot(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3662, 3674, 3672, 3610, 3614, 214} \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {8 \sqrt [4]{-1} a^3 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (a^3 \cot (c+d x)+i a^3\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {8 a^3 (B+i A)}{d \sqrt {\cot (c+d x)}}+\frac {2 i a B (a \cot (c+d x)+i a)^2}{7 d \cot ^{\frac {7}{2}}(c+d x)} \]

[In]

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

[Out]

(8*(-1)^(1/4)*a^3*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (8*a^3*(21*A - (23*I)*B))/(105*d*Cot[c
+ d*x]^(3/2)) + (8*a^3*(I*A + B))/(d*Sqrt[Cot[c + d*x]]) + (((2*I)/7)*a*B*(I*a + a*Cot[c + d*x])^2)/(d*Cot[c
+ d*x]^(7/2)) - (2*(7*A - (11*I)*B)*(I*a^3 + a^3*Cot[c + d*x]))/(35*d*Cot[c + d*x]^(5/2))

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]

Rule 3674

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^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x]
)^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c
+ d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m
- 1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && E
qQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(i a+a \cot (c+d x))^3 (B+A \cot (c+d x))}{\cot ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {(i a+a \cot (c+d x))^2 \left (\frac {1}{2} a (7 i A+11 B)+\frac {1}{2} a (7 A-3 i B) \cot (c+d x)\right )}{\cot ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4}{35} \int \frac {(i a+a \cot (c+d x)) \left (a^2 (21 i A+23 B)+2 a^2 (7 A-6 i B) \cot (c+d x)\right )}{\cot ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4}{35} \int \frac {35 a^3 (i A+B)+35 a^3 (A-i B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {8 a^3 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4}{35} \int \frac {35 a^3 (A-i B)-35 a^3 (i A+B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = -\frac {8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {8 a^3 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {\left (280 a^6 (A-i B)^2\right ) \text {Subst}\left (\int \frac {1}{-35 a^3 (A-i B)-35 a^3 (i A+B) x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {8 \sqrt [4]{-1} a^3 (A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {8 a^3 (21 A-23 i B)}{105 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {8 a^3 (i A+B)}{d \sqrt {\cot (c+d x)}}+\frac {2 i a B (i a+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2 (7 A-11 i B) \left (i a^3+a^3 \cot (c+d x)\right )}{35 d \cot ^{\frac {5}{2}}(c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.76 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 a^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (420 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)} \left (420 (i A+B)-35 (3 A-4 i B) \tan (c+d x)-21 i (A-3 i B) \tan ^2(c+d x)-15 i B \tan ^3(c+d x)\right )\right )}{105 d} \]

[In]

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

[Out]

(2*a^3*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(420*(-1)^(1/4)*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] +
Sqrt[Tan[c + d*x]]*(420*(I*A + B) - 35*(3*A - (4*I)*B)*Tan[c + d*x] - (21*I)*(A - (3*I)*B)*Tan[c + d*x]^2 - (
15*I)*B*Tan[c + d*x]^3)))/(105*d)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {-\frac {2 i A}{5}-\frac {6 B}{5}}{\cot \left (d x +c \right )^{\frac {5}{2}}}+\frac {8 i A +8 B}{\sqrt {\cot \left (d x +c \right )}}+\frac {\frac {8 i B}{3}-2 A}{\cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 i B}{7 \cot \left (d x +c \right )^{\frac {7}{2}}}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i A -4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\)	\(261\)
default	\(\frac {a^{3} \left (\frac {-\frac {2 i A}{5}-\frac {6 B}{5}}{\cot \left (d x +c \right )^{\frac {5}{2}}}+\frac {8 i A +8 B}{\sqrt {\cot \left (d x +c \right )}}+\frac {\frac {8 i B}{3}-2 A}{\cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 i B}{7 \cot \left (d x +c \right )^{\frac {7}{2}}}-\frac {\left (-4 i B +4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\left (-4 i A -4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\)	\(261\)

[In]

int((a+I*a*tan(d*x+c))^3*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a^3/d*(2/5*(-I*A-3*B)/cot(d*x+c)^(5/2)+2*(4*I*A+4*B)/cot(d*x+c)^(1/2)+2/3*(4*I*B-3*A)/cot(d*x+c)^(3/2)-2/7*I*B
/cot(d*x+c)^(7/2)-1/4*(4*A-4*I*B)*2^(1/2)*(ln((1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)-2^(1/2)*co
t(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))-1/4*(-4*I*A-4*B)*
2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2
)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))))

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. 563 vs. \(2 (139) = 278\).

Time = 0.28 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.25 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=-\frac {2 \, {\left (105 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\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}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 105 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\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}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 2 \, {\left ({\left (273 \, A - 319 i \, B\right )} a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 3 \, {\left (133 \, A - 109 i \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 5 \, {\left (21 \, A - 19 i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 3 \, {\left (133 \, A - 129 i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 \, {\left (42 \, A - 41 i \, B\right )} a^{3}\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}}\right )}}{105 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate((a+I*a*tan(d*x+c))^3*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(105*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^6/d^2)*(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(
4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)*log(2*((A - I*B)*a^3*e^(2*I*d*x + 2*I*c) - sqrt(-(-I*A^2 - 2*A
*B + I*B^2)*a^6/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*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^3)) - 105*sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^6/d^2)*(d*e^(8*I*d*x + 8*I*c
) + 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)*log(2*((A - I*B)*a^3*e^(2
*I*d*x + 2*I*c) - sqrt(-(-I*A^2 - 2*A*B + I*B^2)*a^6/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*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^3)) - 2*((273*A - 319*I*B)*a^3*e^
(8*I*d*x + 8*I*c) + 3*(133*A - 109*I*B)*a^3*e^(6*I*d*x + 6*I*c) - 5*(21*A - 19*I*B)*a^3*e^(4*I*d*x + 4*I*c) -
3*(133*A - 129*I*B)*a^3*e^(2*I*d*x + 2*I*c) - 4*(42*A - 41*I*B)*a^3)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*
d*x + 2*I*c) - 1)))/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*
x + 2*I*c) + d)

Sympy [F]

\[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=- i a^{3} \left (\int \frac {i A}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {3 A \tan {\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \frac {A \tan ^{3}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {3 B \tan ^{2}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \frac {B \tan ^{4}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {3 i A \tan ^{2}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \frac {i B \tan {\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {3 i B \tan ^{3}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx\right ) \]

[In]

integrate((a+I*a*tan(d*x+c))**3*(A+B*tan(d*x+c))/cot(d*x+c)**(1/2),x)

[Out]

-I*a**3*(Integral(I*A/sqrt(cot(c + d*x)), x) + Integral(-3*A*tan(c + d*x)/sqrt(cot(c + d*x)), x) + Integral(A*
tan(c + d*x)**3/sqrt(cot(c + d*x)), x) + Integral(-3*B*tan(c + d*x)**2/sqrt(cot(c + d*x)), x) + Integral(B*tan
(c + d*x)**4/sqrt(cot(c + d*x)), x) + Integral(-3*I*A*tan(c + d*x)**2/sqrt(cot(c + d*x)), x) + Integral(I*B*ta
n(c + d*x)/sqrt(cot(c + d*x)), x) + Integral(-3*I*B*tan(c + d*x)**3/sqrt(cot(c + d*x)), x))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.28 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=-\frac {2 \, {\left (15 i \, B a^{3} - \frac {21 \, {\left (-i \, A - 3 \, B\right )} a^{3}}{\tan \left (d x + c\right )} + \frac {35 \, {\left (3 \, A - 4 i \, B\right )} a^{3}}{\tan \left (d x + c\right )^{2}} - \frac {420 \, {\left (i \, A + B\right )} a^{3}}{\tan \left (d x + c\right )^{3}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} - 105 \, {\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3}}{105 \, d} \]

[In]

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

[Out]

-1/105*(2*(15*I*B*a^3 - 21*(-I*A - 3*B)*a^3/tan(d*x + c) + 35*(3*A - 4*I*B)*a^3/tan(d*x + c)^2 - 420*(I*A + B)
*a^3/tan(d*x + c)^3)*tan(d*x + c)^(7/2) - 105*(2*sqrt(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^3)/d

Giac [F]

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

[In]

integrate((a+I*a*tan(d*x+c))^3*(A+B*tan(d*x+c))/cot(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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