\(\int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [35]

Optimal result

Integrand size = 34, antiderivative size = 223 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-8 a^4 (i A+B) x-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d} \]

[Out]

-8*a^4*(I*A+B)*x-8*a^4*(I*A+B)*cot(d*x+c)/d-4*a^4*(A-I*B)*cot(d*x+c)^2/d+1/60*a^4*(93*I*A+92*B)*cot(d*x+c)^3/d
-8*a^4*(A-I*B)*ln(sin(d*x+c))/d-1/6*a*A*cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3/d-1/10*(3*I*A+2*B)*cot(d*x+c)^5*(a^2
+I*a^2*tan(d*x+c))^2/d+1/20*(13*A-12*I*B)*cot(d*x+c)^4*(a^4+I*a^4*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3674, 3672, 3610, 3612, 3556} \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{60 d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}-\frac {8 a^4 (B+i A) \cot (c+d x)}{d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-8 a^4 x (B+i A)-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]

[In]

Int[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-8*a^4*(I*A + B)*x - (8*a^4*(I*A + B)*Cot[c + d*x])/d - (4*a^4*(A - I*B)*Cot[c + d*x]^2)/d + (a^4*((93*I)*A +
92*B)*Cot[c + d*x]^3)/(60*d) - (8*a^4*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^6*(a + I*a*Tan[c + d*
x])^3)/(6*d) - (((3*I)*A + 2*B)*Cot[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^2)/(10*d) + ((13*A - (12*I)*B)*Cot[c
+ d*x]^4*(a^4 + I*a^4*Tan[c + d*x]))/(20*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 ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac {1}{6} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 (3 a (3 i A+2 B)-3 a (A-2 i B) \tan (c+d x)) \, dx \\ & = -\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {1}{30} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \left (-6 a^2 (13 A-12 i B)-6 a^2 (7 i A+8 B) \tan (c+d x)\right ) \, dx \\ & = -\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot ^4(c+d x) (a+i a \tan (c+d x)) \left (-6 a^3 (93 i A+92 B)+6 a^3 (67 A-68 i B) \tan (c+d x)\right ) \, dx \\ & = \frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot ^3(c+d x) \left (960 a^4 (A-i B)+960 a^4 (i A+B) \tan (c+d x)\right ) \, dx \\ & = -\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot ^2(c+d x) \left (960 a^4 (i A+B)-960 a^4 (A-i B) \tan (c+d x)\right ) \, dx \\ & = -\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot (c+d x) \left (-960 a^4 (A-i B)-960 a^4 (i A+B) \tan (c+d x)\right ) \, dx \\ & = -8 a^4 (i A+B) x-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-\left (8 a^4 (A-i B)\right ) \int \cot (c+d x) \, dx \\ & = -8 a^4 (i A+B) x-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.74 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 \left ((13 A-12 i B) (i+\cot (c+d x))^4-4 i (2 A-3 i B) \cot (c+d x) (i+\cot (c+d x))^4-10 A \cot ^2(c+d x) (i+\cot (c+d x))^4+20 (i A+B) \left (-21 \cot (c+d x)+6 i \cot ^2(c+d x)+\cot ^3(c+d x)+24 i (\log (\tan (c+d x))-\log (i+\tan (c+d x)))\right )\right )}{60 d} \]

[In]

Integrate[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(a^4*((13*A - (12*I)*B)*(I + Cot[c + d*x])^4 - (4*I)*(2*A - (3*I)*B)*Cot[c + d*x]*(I + Cot[c + d*x])^4 - 10*A*
Cot[c + d*x]^2*(I + Cot[c + d*x])^4 + 20*(I*A + 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 = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(-\frac {8 a^{4} \left (\left (-\frac {A}{2}+\frac {i B}{2}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (-i B +A \right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {A \left (\cot ^{6}\left (d x +c \right )\right )}{48}+\left (\cot ^{5}\left (d x +c \right )\right ) \left (\frac {i A}{10}+\frac {B}{40}\right )+\left (\cot ^{4}\left (d x +c \right )\right ) \left (\frac {i B}{8}-\frac {7 A}{32}\right )+\left (\cot ^{3}\left (d x +c \right )\right ) \left (-\frac {i A}{3}-\frac {7 B}{24}\right )+\left (\cot ^{2}\left (d x +c \right )\right ) \left (\frac {A}{2}-\frac {i B}{2}\right )+\cot \left (d x +c \right ) \left (i A +B \right )+\left (i A +B \right ) x d \right )}{d}\)	\(143\)
derivativedivides	\(\frac {a^{4} \left (-\frac {4 i A \left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {A \left (\cot ^{6}\left (d x +c \right )\right )}{6}-i B \left (\cot ^{4}\left (d x +c \right )\right )-\frac {B \left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {8 i A \left (\cot ^{3}\left (d x +c \right )\right )}{3}+\frac {7 A \left (\cot ^{4}\left (d x +c \right )\right )}{4}+4 i B \left (\cot ^{2}\left (d x +c \right )\right )+\frac {7 B \left (\cot ^{3}\left (d x +c \right )\right )}{3}-8 i A \cot \left (d x +c \right )-4 A \left (\cot ^{2}\left (d x +c \right )\right )-8 \cot \left (d x +c \right ) B +\frac {\left (-8 i B +8 A \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (8 i A +8 B \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\)	\(174\)
default	\(\frac {a^{4} \left (-\frac {4 i A \left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {A \left (\cot ^{6}\left (d x +c \right )\right )}{6}-i B \left (\cot ^{4}\left (d x +c \right )\right )-\frac {B \left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {8 i A \left (\cot ^{3}\left (d x +c \right )\right )}{3}+\frac {7 A \left (\cot ^{4}\left (d x +c \right )\right )}{4}+4 i B \left (\cot ^{2}\left (d x +c \right )\right )+\frac {7 B \left (\cot ^{3}\left (d x +c \right )\right )}{3}-8 i A \cot \left (d x +c \right )-4 A \left (\cot ^{2}\left (d x +c \right )\right )-8 \cot \left (d x +c \right ) B +\frac {\left (-8 i B +8 A \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (8 i A +8 B \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\)	\(174\)
risch	\(\frac {16 a^{4} B c}{d}+\frac {16 i a^{4} A c}{d}-\frac {4 i a^{4} \left (270 i A \,{\mathrm e}^{10 i \left (d x +c \right )}+210 B \,{\mathrm e}^{10 i \left (d x +c \right )}-855 i A \,{\mathrm e}^{8 i \left (d x +c \right )}-765 B \,{\mathrm e}^{8 i \left (d x +c \right )}+1350 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+1210 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1125 i A \,{\mathrm e}^{4 i \left (d x +c \right )}-1020 B \,{\mathrm e}^{4 i \left (d x +c \right )}+486 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+444 B \,{\mathrm e}^{2 i \left (d x +c \right )}-86 i A -79 B \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {8 A \,a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(221\)
norman	\(\frac {\left (-8 i A \,a^{4}-8 B \,a^{4}\right ) x \left (\tan ^{6}\left (d x +c \right )\right )-\frac {A \,a^{4}}{6 d}+\frac {\left (-4 i B \,a^{4}+7 A \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )}{5 d}-\frac {4 \left (-i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {8 \left (i A \,a^{4}+B \,a^{4}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{d}+\frac {\left (8 i A \,a^{4}+7 B \,a^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}}{\tan \left (d x +c \right )^{6}}-\frac {8 \left (-i B \,a^{4}+A \,a^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {4 \left (-i B \,a^{4}+A \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(229\)

[In]

int(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-8*a^4*((-1/2*A+1/2*I*B)*ln(sec(d*x+c)^2)+(A-I*B)*ln(tan(d*x+c))+1/48*A*cot(d*x+c)^6+cot(d*x+c)^5*(1/10*I*A+1/
40*B)+cot(d*x+c)^4*(1/8*I*B-7/32*A)+cot(d*x+c)^3*(-1/3*I*A-7/24*B)+cot(d*x+c)^2*(1/2*A-1/2*I*B)+cot(d*x+c)*(I*
A+B)+(I*A+B)*x*d)/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.49 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {4 \, {\left (30 \, {\left (9 \, A - 7 i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 \, {\left (19 \, A - 17 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, {\left (135 \, A - 121 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 15 \, {\left (75 \, A - 68 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, {\left (81 \, A - 74 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (86 \, A - 79 i \, B\right )} a^{4} - 30 \, {\left ({\left (A - i \, B\right )} a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, {\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, {\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, {\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

4/15*(30*(9*A - 7*I*B)*a^4*e^(10*I*d*x + 10*I*c) - 45*(19*A - 17*I*B)*a^4*e^(8*I*d*x + 8*I*c) + 10*(135*A - 12
1*I*B)*a^4*e^(6*I*d*x + 6*I*c) - 15*(75*A - 68*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 6*(81*A - 74*I*B)*a^4*e^(2*I*d*x
+ 2*I*c) - (86*A - 79*I*B)*a^4 - 30*((A - I*B)*a^4*e^(12*I*d*x + 12*I*c) - 6*(A - I*B)*a^4*e^(10*I*d*x + 10*I
*c) + 15*(A - I*B)*a^4*e^(8*I*d*x + 8*I*c) - 20*(A - I*B)*a^4*e^(6*I*d*x + 6*I*c) + 15*(A - I*B)*a^4*e^(4*I*d*
x + 4*I*c) - 6*(A - I*B)*a^4*e^(2*I*d*x + 2*I*c) + (A - I*B)*a^4)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(12*I*d*x
+ 12*I*c) - 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^(8*I*d*x + 8*I*c) - 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x
+ 4*I*c) - 6*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [A] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.56 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=- \frac {8 a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 344 A a^{4} + 316 i B a^{4} + \left (1944 A a^{4} e^{2 i c} - 1776 i B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 4500 A a^{4} e^{4 i c} + 4080 i B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (5400 A a^{4} e^{6 i c} - 4840 i B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 3420 A a^{4} e^{8 i c} + 3060 i B a^{4} e^{8 i c}\right ) e^{8 i d x} + \left (1080 A a^{4} e^{10 i c} - 840 i B a^{4} e^{10 i c}\right ) e^{10 i d x}}{15 d e^{12 i c} e^{12 i d x} - 90 d e^{10 i c} e^{10 i d x} + 225 d e^{8 i c} e^{8 i d x} - 300 d e^{6 i c} e^{6 i d x} + 225 d e^{4 i c} e^{4 i d x} - 90 d e^{2 i c} e^{2 i d x} + 15 d} \]

[In]

integrate(cot(d*x+c)**7*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

-8*a**4*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-344*A*a**4 + 316*I*B*a**4 + (1944*A*a**4*exp(2*I*c) -
1776*I*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (-4500*A*a**4*exp(4*I*c) + 4080*I*B*a**4*exp(4*I*c))*exp(4*I*d*x) + (
5400*A*a**4*exp(6*I*c) - 4840*I*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (-3420*A*a**4*exp(8*I*c) + 3060*I*B*a**4*exp
(8*I*c))*exp(8*I*d*x) + (1080*A*a**4*exp(10*I*c) - 840*I*B*a**4*exp(10*I*c))*exp(10*I*d*x))/(15*d*exp(12*I*c)*
exp(12*I*d*x) - 90*d*exp(10*I*c)*exp(10*I*d*x) + 225*d*exp(8*I*c)*exp(8*I*d*x) - 300*d*exp(6*I*c)*exp(6*I*d*x)
+ 225*d*exp(4*I*c)*exp(4*I*d*x) - 90*d*exp(2*I*c)*exp(2*I*d*x) + 15*d)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.77 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {480 \, {\left (d x + c\right )} {\left (i \, A + B\right )} a^{4} - 240 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac {480 \, {\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )^{5} - 240 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} + 20 \, {\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} + 15 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 12 \, {\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) - 10 \, A a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \]

[In]

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(480*(d*x + c)*(I*A + B)*a^4 - 240*(A - I*B)*a^4*log(tan(d*x + c)^2 + 1) + 480*(A - I*B)*a^4*log(tan(d*x
+ c)) - (480*(-I*A - B)*a^4*tan(d*x + c)^5 - 240*(A - I*B)*a^4*tan(d*x + c)^4 + 20*(8*I*A + 7*B)*a^4*tan(d*x
+ c)^3 + 15*(7*A - 4*I*B)*a^4*tan(d*x + c)^2 + 12*(-4*I*A - B)*a^4*tan(d*x + c) - 10*A*a^4)/tan(d*x + c)^6)/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. 459 vs. \(2 (195) = 390\).

Time = 1.29 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.06 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {5 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 620 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2835 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2400 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10080 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 30720 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 15360 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {37632 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 37632 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2835 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2400 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 620 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/1920*(5*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 12*B*a^4*tan(1/2*d*x + 1/2*c)^5
- 240*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 120*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 880*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 +
620*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 2835*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 2400*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 -
10080*I*A*a^4*tan(1/2*d*x + 1/2*c) - 9480*B*a^4*tan(1/2*d*x + 1/2*c) - 30720*(A*a^4 - I*B*a^4)*log(tan(1/2*d*x
+ 1/2*c) + I) + 15360*(A*a^4 - I*B*a^4)*log(tan(1/2*d*x + 1/2*c)) - (37632*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 376
32*I*B*a^4*tan(1/2*d*x + 1/2*c)^6 - 10080*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 9480*B*a^4*tan(1/2*d*x + 1/2*c)^5 -
2835*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 2400*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 880*I*A*a^4*tan(1/2*d*x + 1/2*c)^3
+ 620*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 240*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 120*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 - 4
8*I*A*a^4*tan(1/2*d*x + 1/2*c) - 12*B*a^4*tan(1/2*d*x + 1/2*c) - 5*A*a^4)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 9.45 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.73 \[ \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (4\,A\,a^4-B\,a^4\,4{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A\,a^4}{4}-B\,a^4\,1{}\mathrm {i}\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (8\,B\,a^4+A\,a^4\,8{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {7\,B\,a^4}{3}+\frac {A\,a^4\,8{}\mathrm {i}}{3}\right )+\frac {A\,a^4}{6}+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{5}+\frac {A\,a^4\,4{}\mathrm {i}}{5}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^6}-\frac {16\,a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{d} \]

[In]

int(cot(c + d*x)^7*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

- (tan(c + d*x)^4*(4*A*a^4 - B*a^4*4i) - tan(c + d*x)^2*((7*A*a^4)/4 - B*a^4*1i) + tan(c + d*x)^5*(A*a^4*8i +
8*B*a^4) - tan(c + d*x)^3*((A*a^4*8i)/3 + (7*B*a^4)/3) + (A*a^4)/6 + tan(c + d*x)*((A*a^4*4i)/5 + (B*a^4)/5))/
(d*tan(c + d*x)^6) - (16*a^4*atan(2*tan(c + d*x) + 1i)*(A*1i + B))/d