\(\int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [34]

Optimal result

Integrand size = 34, antiderivative size = 200 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-8 a^4 (A-i B) x-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}+\frac {8 a^4 (i A+B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d} \]

[Out]

-8*a^4*(A-I*B)*x-8*a^4*(A-I*B)*cot(d*x+c)/d+1/60*a^4*(148*I*A+145*B)*cot(d*x+c)^2/d+8*a^4*(I*A+B)*ln(sin(d*x+c
))/d-1/5*a*A*cot(d*x+c)^5*(a+I*a*tan(d*x+c))^3/d-1/20*(8*I*A+5*B)*cot(d*x+c)^4*(a^2+I*a^2*tan(d*x+c))^2/d+1/30
*(28*A-25*I*B)*cot(d*x+c)^3*(a^4+I*a^4*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3674, 3672, 3610, 3612, 3556} \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {8 a^4 (B+i A) \log (\sin (c+d x))}{d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}-8 a^4 x (A-i B)-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d} \]

[In]

Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-8*a^4*(A - I*B)*x - (8*a^4*(A - I*B)*Cot[c + d*x])/d + (a^4*((148*I)*A + 145*B)*Cot[c + d*x]^2)/(60*d) + (8*a
^4*(I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3)/(5*d) - (((8*I)*A + 5*B)*Cot
[c + d*x]^4*(a^2 + I*a^2*Tan[c + d*x])^2)/(20*d) + ((28*A - (25*I)*B)*Cot[c + d*x]^3*(a^4 + I*a^4*Tan[c + d*x]
))/(30*d)

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 3612

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

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}& = -\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 (a (8 i A+5 B)-a (2 A-5 i B) \tan (c+d x)) \, dx \\ & = -\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {1}{20} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \left (-2 a^2 (28 A-25 i B)-6 a^2 (4 i A+5 B) \tan (c+d x)\right ) \, dx \\ & = -\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac {1}{60} \int \cot ^3(c+d x) (a+i a \tan (c+d x)) \left (-2 a^3 (148 i A+145 B)+2 a^3 (92 A-95 i B) \tan (c+d x)\right ) \, dx \\ & = \frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac {1}{60} \int \cot ^2(c+d x) \left (480 a^4 (A-i B)+480 a^4 (i A+B) \tan (c+d x)\right ) \, dx \\ & = -\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac {1}{60} \int \cot (c+d x) \left (480 a^4 (i A+B)-480 a^4 (A-i B) \tan (c+d x)\right ) \, dx \\ & = -8 a^4 (A-i B) x-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\left (8 a^4 (i A+B)\right ) \int \cot (c+d x) \, dx \\ & = -8 a^4 (A-i B) x-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}+\frac {8 a^4 (i A+B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.60 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 \left (-3 i (4 A-5 i B) (i+\cot (c+d x))^4-12 A \cot (c+d x) (i+\cot (c+d x))^4+20 (A-i B) \left (-21 \cot (c+d x)+6 i \cot ^2(c+d x)+\cot ^3(c+d x)+24 i (\log (\tan (c+d x))-\log (i+\tan (c+d x)))\right )\right )}{60 d} \]

[In]

Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(a^4*((-3*I)*(4*A - (5*I)*B)*(I + Cot[c + d*x])^4 - 12*A*Cot[c + d*x]*(I + Cot[c + d*x])^4 + 20*(A - I*B)*(-21
*Cot[c + d*x] + (6*I)*Cot[c + d*x]^2 + Cot[c + d*x]^3 + (24*I)*(Log[Tan[c + d*x]] - Log[I + Tan[c + d*x]]))))/
(60*d)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(\frac {8 a^{4} \left (\left (-\frac {i A}{2}-\frac {B}{2}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (i A +B \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \left (\cot ^{5}\left (d x +c \right )\right )}{40}+\left (\cot ^{4}\left (d x +c \right )\right ) \left (-\frac {i A}{8}-\frac {B}{32}\right )+\left (\cot ^{3}\left (d x +c \right )\right ) \left (-\frac {i B}{6}+\frac {7 A}{24}\right )+\left (\cot ^{2}\left (d x +c \right )\right ) \left (\frac {i A}{2}+\frac {7 B}{16}\right )+\left (i B -A \right ) \cot \left (d x +c \right )+x d \left (i B -A \right )\right )}{d}\)	\(130\)
derivativedivides	\(\frac {a^{4} \left (-i A \left (\cot ^{4}\left (d x +c \right )\right )-\frac {A \left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {4 i B \left (\cot ^{3}\left (d x +c \right )\right )}{3}-\frac {B \left (\cot ^{4}\left (d x +c \right )\right )}{4}+4 i A \left (\cot ^{2}\left (d x +c \right )\right )+\frac {7 A \left (\cot ^{3}\left (d x +c \right )\right )}{3}+8 i B \cot \left (d x +c \right )+\frac {7 B \left (\cot ^{2}\left (d x +c \right )\right )}{2}-8 A \cot \left (d x +c \right )+\frac {\left (-8 i A -8 B \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i B +8 A \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\)	\(151\)
default	\(\frac {a^{4} \left (-i A \left (\cot ^{4}\left (d x +c \right )\right )-\frac {A \left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {4 i B \left (\cot ^{3}\left (d x +c \right )\right )}{3}-\frac {B \left (\cot ^{4}\left (d x +c \right )\right )}{4}+4 i A \left (\cot ^{2}\left (d x +c \right )\right )+\frac {7 A \left (\cot ^{3}\left (d x +c \right )\right )}{3}+8 i B \cot \left (d x +c \right )+\frac {7 B \left (\cot ^{2}\left (d x +c \right )\right )}{2}-8 A \cot \left (d x +c \right )+\frac {\left (-8 i A -8 B \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i B +8 A \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\)	\(151\)
risch	\(-\frac {16 i a^{4} B c}{d}+\frac {16 a^{4} A c}{d}-\frac {4 a^{4} \left (210 i A \,{\mathrm e}^{8 i \left (d x +c \right )}+150 B \,{\mathrm e}^{8 i \left (d x +c \right )}-555 i A \,{\mathrm e}^{6 i \left (d x +c \right )}-465 B \,{\mathrm e}^{6 i \left (d x +c \right )}+655 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+565 B \,{\mathrm e}^{4 i \left (d x +c \right )}-365 i A \,{\mathrm e}^{2 i \left (d x +c \right )}-320 B \,{\mathrm e}^{2 i \left (d x +c \right )}+79 i A +70 B \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{d}\)	\(195\)
norman	\(\frac {\left (8 i B \,a^{4}-8 A \,a^{4}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {A \,a^{4}}{5 d}+\frac {\left (-4 i B \,a^{4}+7 A \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )}{4 d}-\frac {8 \left (-i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {\left (8 i A \,a^{4}+7 B \,a^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {8 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {4 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(203\)

[In]

int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

8*a^4*((-1/2*I*A-1/2*B)*ln(sec(d*x+c)^2)+(I*A+B)*ln(tan(d*x+c))-1/40*A*cot(d*x+c)^5+cot(d*x+c)^4*(-1/8*I*A-1/3
2*B)+cot(d*x+c)^3*(-1/6*I*B+7/24*A)+cot(d*x+c)^2*(1/2*I*A+7/16*B)+(-A+I*B)*cot(d*x+c)+x*d*(-A+I*B))/d

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.44 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (30 \, {\left (7 i \, A + 5 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 15 \, {\left (-37 i \, A - 31 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \, {\left (131 i \, A + 113 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, {\left (-73 i \, A - 64 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (79 i \, A + 70 \, B\right )} a^{4} + 30 \, {\left ({\left (-i \, A - B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, {\left (i \, A + B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, {\left (-i \, A - B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, {\left (i \, A + B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, {\left (-i \, A - B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A + B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-4/15*(30*(7*I*A + 5*B)*a^4*e^(8*I*d*x + 8*I*c) + 15*(-37*I*A - 31*B)*a^4*e^(6*I*d*x + 6*I*c) + 5*(131*I*A + 1
13*B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(-73*I*A - 64*B)*a^4*e^(2*I*d*x + 2*I*c) + (79*I*A + 70*B)*a^4 + 30*((-I*A -
B)*a^4*e^(10*I*d*x + 10*I*c) + 5*(I*A + B)*a^4*e^(8*I*d*x + 8*I*c) + 10*(-I*A - B)*a^4*e^(6*I*d*x + 6*I*c) +
10*(I*A + B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(-I*A - B)*a^4*e^(2*I*d*x + 2*I*c) + (I*A + B)*a^4)*log(e^(2*I*d*x +
2*I*c) - 1))/(d*e^(10*I*d*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x +
4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) - d)

Sympy [A] (verification not implemented)

Time = 2.52 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.48 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {8 i a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 316 i A a^{4} - 280 B a^{4} + \left (1460 i A a^{4} e^{2 i c} + 1280 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 2620 i A a^{4} e^{4 i c} - 2260 B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (2220 i A a^{4} e^{6 i c} + 1860 B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 840 i A a^{4} e^{8 i c} - 600 B a^{4} e^{8 i c}\right ) e^{8 i d x}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \]

[In]

integrate(cot(d*x+c)**6*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

8*I*a**4*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-316*I*A*a**4 - 280*B*a**4 + (1460*I*A*a**4*exp(2*I*c)
+ 1280*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (-2620*I*A*a**4*exp(4*I*c) - 2260*B*a**4*exp(4*I*c))*exp(4*I*d*x) +
(2220*I*A*a**4*exp(6*I*c) + 1860*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (-840*I*A*a**4*exp(8*I*c) - 600*B*a**4*exp(
8*I*c))*exp(8*I*d*x))/(15*d*exp(10*I*c)*exp(10*I*d*x) - 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*
I*d*x) - 150*d*exp(4*I*c)*exp(4*I*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.76 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {480 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} + 240 \, {\left (i \, A + B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, {\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {480 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 30 \, {\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 20 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 15 \, {\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) + 12 \, A a^{4}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(480*(d*x + c)*(A - I*B)*a^4 + 240*(I*A + B)*a^4*log(tan(d*x + c)^2 + 1) + 480*(-I*A - B)*a^4*log(tan(d*
x + c)) + (480*(A - I*B)*a^4*tan(d*x + c)^4 - 30*(8*I*A + 7*B)*a^4*tan(d*x + c)^3 - 20*(7*A - 4*I*B)*a^4*tan(d
*x + c)^2 - 15*(-4*I*A - B)*a^4*tan(d*x + c) + 12*A*a^4)/tan(d*x + c)^5)/d

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (174) = 348\).

Time = 1.22 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.96 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 310 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 900 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4740 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4320 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15360 \, {\left (i \, A a^{4} + B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 7680 \, {\left (-i \, A a^{4} - B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {-17536 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 17536 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4740 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4320 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1200 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 900 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 310 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 160 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/960*(6*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 60*I*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 15*B*a^4*tan(1/2*d*x + 1/2*c)^4 -
310*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 160*I*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 1200*I*A*a^4*tan(1/2*d*x + 1/2*c)^2 +
900*B*a^4*tan(1/2*d*x + 1/2*c)^2 + 4740*A*a^4*tan(1/2*d*x + 1/2*c) - 4320*I*B*a^4*tan(1/2*d*x + 1/2*c) - 15360
*(I*A*a^4 + B*a^4)*log(tan(1/2*d*x + 1/2*c) + I) - 7680*(-I*A*a^4 - B*a^4)*log(tan(1/2*d*x + 1/2*c)) + (-17536
*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 17536*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 4740*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 432
0*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 1200*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 900*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 31
0*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 160*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 - 60*I*A*a^4*tan(1/2*d*x + 1/2*c) - 15*B*a
^4*tan(1/2*d*x + 1/2*c) - 6*A*a^4)/tan(1/2*d*x + 1/2*c)^5)/d

Mupad [B] (verification not implemented)

Time = 7.94 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.70 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {\frac {A\,a^4}{5}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A\,a^4}{3}-\frac {B\,a^4\,4{}\mathrm {i}}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (8\,A\,a^4-B\,a^4\,8{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {7\,B\,a^4}{2}+A\,a^4\,4{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{4}+A\,a^4\,1{}\mathrm {i}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5}+\frac {a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,16{}\mathrm {i}}{d} \]

[In]

int(cot(c + d*x)^6*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

(a^4*atan(2*tan(c + d*x) + 1i)*(A*1i + B)*16i)/d - (tan(c + d*x)^4*(8*A*a^4 - B*a^4*8i) - tan(c + d*x)^2*((7*A
*a^4)/3 - (B*a^4*4i)/3) - tan(c + d*x)^3*(A*a^4*4i + (7*B*a^4)/2) + (A*a^4)/5 + tan(c + d*x)*(A*a^4*1i + (B*a^
4)/4))/(d*tan(c + d*x)^5)