Integrand size = 34, antiderivative size = 163 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=8 a^4 (A-i B) x-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {a^4 (8 i A+7 B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {(2 i A+B) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d} \] Output:
8*a^4*(A-I*B)*x-a^4*B*ln(cos(d*x+c))/d-a^4*(8*I*A+7*B)*ln(sin(d*x+c))/d-1/ 3*a*A*cot(d*x+c)^3*(a+I*a*tan(d*x+c))^3/d-1/2*(2*I*A+B)*cot(d*x+c)^2*(a^2+ I*a^2*tan(d*x+c))^2/d+(4*A-3*I*B)*cot(d*x+c)*(a^4+I*a^4*tan(d*x+c))/d
Time = 0.72 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.88 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=a^4 \left (\frac {7 A \cot (c+d x)}{d}-\frac {4 i B \cot (c+d x)}{d}-\frac {2 i A \cot ^2(c+d x)}{d}-\frac {B \cot ^2(c+d x)}{2 d}-\frac {A \cot ^3(c+d x)}{3 d}-\frac {8 i A \log (\tan (c+d x))}{d}-\frac {7 B \log (\tan (c+d x))}{d}+\frac {8 i A \log (i+\tan (c+d x))}{d}+\frac {8 B \log (i+\tan (c+d x))}{d}\right ) \] Input:
Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]
Output:
a^4*((7*A*Cot[c + d*x])/d - ((4*I)*B*Cot[c + d*x])/d - ((2*I)*A*Cot[c + d* x]^2)/d - (B*Cot[c + d*x]^2)/(2*d) - (A*Cot[c + d*x]^3)/(3*d) - ((8*I)*A*L og[Tan[c + d*x]])/d - (7*B*Log[Tan[c + d*x]])/d + ((8*I)*A*Log[I + Tan[c + d*x]])/d + (8*B*Log[I + Tan[c + d*x]])/d)
Time = 1.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 4076, 27, 3042, 4076, 27, 3042, 4076, 3042, 4072, 3042, 3956, 4014, 3042, 25, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4 (A+B \tan (c+d x))}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{3} \int 3 \cot ^3(c+d x) (i \tan (c+d x) a+a)^3 (a (2 i A+B)+i a B \tan (c+d x))dx-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \cot ^3(c+d x) (i \tan (c+d x) a+a)^3 (a (2 i A+B)+i a B \tan (c+d x))dx-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(i \tan (c+d x) a+a)^3 (a (2 i A+B)+i a B \tan (c+d x))}{\tan (c+d x)^3}dx-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \frac {1}{2} \int -2 \cot ^2(c+d x) (i \tan (c+d x) a+a)^2 \left ((4 A-3 i B) a^2+B \tan (c+d x) a^2\right )dx-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \cot ^2(c+d x) (i \tan (c+d x) a+a)^2 \left ((4 A-3 i B) a^2+B \tan (c+d x) a^2\right )dx-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {(i \tan (c+d x) a+a)^2 \left ((4 A-3 i B) a^2+B \tan (c+d x) a^2\right )}{\tan (c+d x)^2}dx-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle -\int \cot (c+d x) (i \tan (c+d x) a+a) \left ((8 i A+7 B) a^3+i B \tan (c+d x) a^3\right )dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {(i \tan (c+d x) a+a) \left ((8 i A+7 B) a^3+i B \tan (c+d x) a^3\right )}{\tan (c+d x)}dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4072 |
| \(\displaystyle -\int \cot (c+d x) \left (a^4 (8 i A+7 B)-8 a^4 (A-i B) \tan (c+d x)\right )dx+a^4 B \int \tan (c+d x)dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a^4 (8 i A+7 B)-8 a^4 (A-i B) \tan (c+d x)}{\tan (c+d x)}dx+a^4 B \int \tan (c+d x)dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\int \frac {a^4 (8 i A+7 B)-8 a^4 (A-i B) \tan (c+d x)}{\tan (c+d x)}dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle -a^4 (7 B+8 i A) \int \cot (c+d x)dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^4 (7 B+8 i A) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^4 (7 B+8 i A) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {a^4 (7 B+8 i A) \log (-\sin (c+d x))}{d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
Input:
Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]
Output:
8*a^4*(A - I*B)*x - (a^4*B*Log[Cos[c + d*x]])/d - (a^4*((8*I)*A + 7*B)*Log [-Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(3*d) - (((2*I)*A + B)*Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) + ((4*A - (3*I)*B)*Cot[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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] + Simp[(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] && N eQ[a*c + b*d, 0]
Int[(((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_ .)*(x_)]))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B*(d/ b) Int[Tan[e + f*x], x], x] + Simp[1/b Int[Simp[A*b*c + (A*b*d + B*(b*c - a*d))*Tan[e + f*x], x]/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d , e, f, A, B}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-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] - Simp[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] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Time = 0.72 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {8 \left (\frac {\left (i A +B \right ) \ln \left (\sec \left (d x +c \right )^{2}\right )}{2}-\left (i A +\frac {7 B}{8}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \cot \left (d x +c \right )^{3}}{24}-\frac {\cot \left (d x +c \right )^{2} \left (i A +\frac {B}{4}\right )}{4}+\frac {\left (-i B +\frac {7 A}{4}\right ) \cot \left (d x +c \right )}{2}+\left (-i B +A \right ) x d \right ) a^{4}}{d}\) | \(98\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\left (8 i A +8 B \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-8 i B +8 A \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{3 \tan \left (d x +c \right )^{3}}+\left (-8 i A -7 B \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {4 i A +B}{2 \tan \left (d x +c \right )^{2}}-\frac {4 i B -7 A}{\tan \left (d x +c \right )}\right )}{d}\) | \(107\) |
| default | \(\frac {a^{4} \left (\frac {\left (8 i A +8 B \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-8 i B +8 A \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{3 \tan \left (d x +c \right )^{3}}+\left (-8 i A -7 B \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {4 i A +B}{2 \tan \left (d x +c \right )^{2}}-\frac {4 i B -7 A}{\tan \left (d x +c \right )}\right )}{d}\) | \(107\) |
| norman | \(\frac {\frac {\left (-4 i B \,a^{4}+7 A \,a^{4}\right ) \tan \left (d x +c \right )^{2}}{d}+\left (-8 i B \,a^{4}+8 A \,a^{4}\right ) x \tan \left (d x +c \right )^{3}-\frac {A \,a^{4}}{3 d}-\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}+\frac {4 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}-\frac {\left (8 i A \,a^{4}+7 B \,a^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(150\) |
| risch | \(\frac {16 i a^{4} B c}{d}-\frac {16 a^{4} A c}{d}+\frac {2 a^{4} \left (36 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+15 B \,{\mathrm e}^{4 i \left (d x +c \right )}-54 i A \,{\mathrm e}^{2 i \left (d x +c \right )}-27 B \,{\mathrm e}^{2 i \left (d x +c \right )}+22 i A +12 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {7 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}\) | \(166\) |
Input:
int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVER BOSE)
Output:
8*(1/2*(I*A+B)*ln(sec(d*x+c)^2)-(I*A+7/8*B)*ln(tan(d*x+c))-1/24*A*cot(d*x+ c)^3-1/4*cot(d*x+c)^2*(I*A+1/4*B)+1/2*(-I*B+7/4*A)*cot(d*x+c)+(A-I*B)*x*d) *a^4/d
Time = 0.10 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.53 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {6 \, {\left (-12 i \, A - 5 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 54 \, {\left (2 i \, A + B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, {\left (-11 i \, A - 6 \, B\right )} a^{4} + 3 \, {\left (B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - B a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 \, {\left ({\left (8 i \, A + 7 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (-8 i \, A - 7 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, {\left (8 i \, A + 7 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-8 i \, A - 7 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \] Input:
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm= "fricas")
Output:
-1/3*(6*(-12*I*A - 5*B)*a^4*e^(4*I*d*x + 4*I*c) + 54*(2*I*A + B)*a^4*e^(2* I*d*x + 2*I*c) + 4*(-11*I*A - 6*B)*a^4 + 3*(B*a^4*e^(6*I*d*x + 6*I*c) - 3* B*a^4*e^(4*I*d*x + 4*I*c) + 3*B*a^4*e^(2*I*d*x + 2*I*c) - B*a^4)*log(e^(2* I*d*x + 2*I*c) + 1) + 3*((8*I*A + 7*B)*a^4*e^(6*I*d*x + 6*I*c) + 3*(-8*I*A - 7*B)*a^4*e^(4*I*d*x + 4*I*c) + 3*(8*I*A + 7*B)*a^4*e^(2*I*d*x + 2*I*c) + (-8*I*A - 7*B)*a^4)*log(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)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (143) = 286\).
Time = 2.71 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.79 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=- \frac {B a^{4} \log {\left (\frac {4 A a^{4} - 3 i B a^{4}}{4 A a^{4} e^{2 i c} - 3 i B a^{4} e^{2 i c}} + e^{2 i d x} \right )}}{d} - \frac {i a^{4} \cdot \left (8 A - 7 i B\right ) \log {\left (e^{2 i d x} + \frac {4 A a^{4} - 4 i B a^{4} - a^{4} \cdot \left (8 A - 7 i B\right )}{4 A a^{4} e^{2 i c} - 3 i B a^{4} e^{2 i c}} \right )}}{d} + \frac {44 i A a^{4} + 24 B a^{4} + \left (- 108 i A a^{4} e^{2 i c} - 54 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (72 i A a^{4} e^{4 i c} + 30 B a^{4} e^{4 i c}\right ) e^{4 i d x}}{3 d e^{6 i c} e^{6 i d x} - 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} - 3 d} \] Input:
integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)
Output:
-B*a**4*log((4*A*a**4 - 3*I*B*a**4)/(4*A*a**4*exp(2*I*c) - 3*I*B*a**4*exp( 2*I*c)) + exp(2*I*d*x))/d - I*a**4*(8*A - 7*I*B)*log(exp(2*I*d*x) + (4*A*a **4 - 4*I*B*a**4 - a**4*(8*A - 7*I*B))/(4*A*a**4*exp(2*I*c) - 3*I*B*a**4*e xp(2*I*c)))/d + (44*I*A*a**4 + 24*B*a**4 + (-108*I*A*a**4*exp(2*I*c) - 54* B*a**4*exp(2*I*c))*exp(2*I*d*x) + (72*I*A*a**4*exp(4*I*c) + 30*B*a**4*exp( 4*I*c))*exp(4*I*d*x))/(3*d*exp(6*I*c)*exp(6*I*d*x) - 9*d*exp(4*I*c)*exp(4* I*d*x) + 9*d*exp(2*I*c)*exp(2*I*d*x) - 3*d)
Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.72 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {48 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} - 24 \, {\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, {\left (-8 i \, A - 7 \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 3 \, {\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) - 2 \, A a^{4}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm= "maxima")
Output:
1/6*(48*(d*x + c)*(A - I*B)*a^4 - 24*(-I*A - B)*a^4*log(tan(d*x + c)^2 + 1 ) + 6*(-8*I*A - 7*B)*a^4*log(tan(d*x + c)) + (6*(7*A - 4*I*B)*a^4*tan(d*x + c)^2 + 3*(-4*I*A - B)*a^4*tan(d*x + c) - 2*A*a^4)/tan(d*x + c)^3)/d
Time = 0.52 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.71 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {8 \, {\left (-i \, A a^{4} - B a^{4}\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{d} + \frac {{\left (-8 i \, A a^{4} - 7 \, B a^{4}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {2 \, A a^{4} - 6 \, {\left (7 \, A a^{4} - 4 i \, B a^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (4 i \, A a^{4} + B a^{4}\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \] Input:
integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm= "giac")
Output:
-8*(-I*A*a^4 - B*a^4)*log(tan(d*x + c) + I)/d + (-8*I*A*a^4 - 7*B*a^4)*log (abs(tan(d*x + c)))/d - 1/6*(2*A*a^4 - 6*(7*A*a^4 - 4*I*B*a^4)*tan(d*x + c )^2 + 3*(4*I*A*a^4 + B*a^4)*tan(d*x + c))/(d*tan(d*x + c)^3)
Time = 3.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.69 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {\frac {A\,a^4}{3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (7\,A\,a^4-B\,a^4\,4{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{2}+A\,a^4\,2{}\mathrm {i}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3}-\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (7\,B+A\,8{}\mathrm {i}\right )}{d}+\frac {8\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{d} \] Input:
int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)
Output:
(8*a^4*log(tan(c + d*x) + 1i)*(A*1i + B))/d - (a^4*log(tan(c + d*x))*(A*8i + 7*B))/d - ((A*a^4)/3 - tan(c + d*x)^2*(7*A*a^4 - B*a^4*4i) + tan(c + d* x)*(A*a^4*2i + (B*a^4)/2))/(d*tan(c + d*x)^3)
Time = 0.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.66 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^{4} \left (88 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a -48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b i -4 \cos \left (d x +c \right ) a +96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{3} a i +96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{3} b -12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{3} b -12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3} b -96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3} a i -84 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3} b +96 \sin \left (d x +c \right )^{3} a d x +12 \sin \left (d x +c \right )^{3} a i -96 \sin \left (d x +c \right )^{3} b d i x +3 \sin \left (d x +c \right )^{3} b -24 \sin \left (d x +c \right ) a i -6 \sin \left (d x +c \right ) b \right )}{12 \sin \left (d x +c \right )^{3} d} \] Input:
int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)
Output:
(a**4*(88*cos(c + d*x)*sin(c + d*x)**2*a - 48*cos(c + d*x)*sin(c + d*x)**2 *b*i - 4*cos(c + d*x)*a + 96*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3* a*i + 96*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*b - 12*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**3*b - 12*log(tan((c + d*x)/2) + 1)*sin(c + d*x) **3*b - 96*log(tan((c + d*x)/2))*sin(c + d*x)**3*a*i - 84*log(tan((c + d*x )/2))*sin(c + d*x)**3*b + 96*sin(c + d*x)**3*a*d*x + 12*sin(c + d*x)**3*a* i - 96*sin(c + d*x)**3*b*d*i*x + 3*sin(c + d*x)**3*b - 24*sin(c + d*x)*a*i - 6*sin(c + d*x)*b))/(12*sin(c + d*x)**3*d)