Integrand size = 24, antiderivative size = 103 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=8 a^4 x+\frac {4 a^4 \cot (c+d x)}{d}-\frac {8 i a^4 \log (\sin (c+d x))}{d}-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]
8*a^4*x+4*a^4*cot(d*x+c)/d-8*I*a^4*ln(sin(d*x+c))/d-1/3*a*cot(d*x+c)^3*(a+ I*a*tan(d*x+c))^3/d-I*cot(d*x+c)^2*(a^2+I*a^2*tan(d*x+c))^2/d
Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=a^4 \left (\frac {7 \cot (c+d x)}{d}-\frac {2 i \cot ^2(c+d x)}{d}-\frac {\cot ^3(c+d x)}{3 d}-\frac {8 i \log (\tan (c+d x))}{d}+\frac {8 i \log (i+\tan (c+d x))}{d}\right ) \]
a^4*((7*Cot[c + d*x])/d - ((2*I)*Cot[c + d*x]^2)/d - Cot[c + d*x]^3/(3*d) - ((8*I)*Log[Tan[c + d*x]])/d + ((8*I)*Log[I + Tan[c + d*x]])/d)
Time = 0.66 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4028, 3042, 4028, 3042, 4025, 27, 3042, 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.
| \(\displaystyle \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 4028 |
| \(\displaystyle 2 i a \int \cot ^3(c+d x) (i \tan (c+d x) a+a)^3dx-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 i a \int \frac {(i \tan (c+d x) a+a)^3}{\tan (c+d x)^3}dx-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4028 |
| \(\displaystyle 2 i a \left (2 i a \int \cot ^2(c+d x) (i \tan (c+d x) a+a)^2dx-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 i a \left (2 i a \int \frac {(i \tan (c+d x) a+a)^2}{\tan (c+d x)^2}dx-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+\int 2 \cot (c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+2 \int \cot (c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+2 \int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)}dx\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+2 \left (-a^2 x+i a^2 \int \cot (c+d x)dx\right )\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+2 \left (-a^2 x+i a^2 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx\right )\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+2 \left (a^2 (-x)-i a^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx\right )\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle 2 i a \left (2 i a \left (-\frac {a^2 \cot (c+d x)}{d}+2 \left (-a^2 x+\frac {i a^2 \log (-\sin (c+d x))}{d}\right )\right )-\frac {a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}\right )-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\) |
-1/3*(a*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d + (2*I)*a*((2*I)*a*(-(( a^2*Cot[c + d*x])/d) + 2*(-(a^2*x) + (I*a^2*Log[-Sin[c + d*x]])/d)) - (a*C ot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2)/(2*d))
3.1.41.3.1 Defintions of rubi rules used
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_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*b*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m - 1)*(a*c - b*d))), x] + Simp[2*(a^2/(a*c - b*d)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0 ] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2]
Time = 0.57 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.61
| method | result | size |
| parallelrisch | \(-\frac {a^{4} \left (\cot ^{3}\left (d x +c \right )+6 i \left (\cot ^{2}\left (d x +c \right )\right )+24 i \ln \left (\tan \left (d x +c \right )\right )-12 i \ln \left (\sec ^{2}\left (d x +c \right )\right )-24 d x -21 \cot \left (d x +c \right )\right )}{3 d}\) | \(63\) |
| derivativedivides | \(\frac {a^{4} \left (-\frac {1}{3 \tan \left (d x +c \right )^{3}}-8 i \ln \left (\tan \left (d x +c \right )\right )-\frac {2 i}{\tan \left (d x +c \right )^{2}}+\frac {7}{\tan \left (d x +c \right )}+4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(73\) |
| default | \(\frac {a^{4} \left (-\frac {1}{3 \tan \left (d x +c \right )^{3}}-8 i \ln \left (\tan \left (d x +c \right )\right )-\frac {2 i}{\tan \left (d x +c \right )^{2}}+\frac {7}{\tan \left (d x +c \right )}+4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(73\) |
| risch | \(-\frac {16 a^{4} c}{d}+\frac {4 i a^{4} \left (18 \,{\mathrm e}^{4 i \left (d x +c \right )}-27 \,{\mathrm e}^{2 i \left (d x +c \right )}+11\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(78\) |
| norman | \(\frac {-\frac {a^{4}}{3 d}+8 a^{4} x \left (\tan ^{3}\left (d x +c \right )\right )+\frac {7 a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {2 i a^{4} \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{3}}-\frac {8 i a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {4 i a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(101\) |
-1/3*a^4*(cot(d*x+c)^3+6*I*cot(d*x+c)^2+24*I*ln(tan(d*x+c))-12*I*ln(sec(d* x+c)^2)-24*d*x-21*cot(d*x+c))/d
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.35 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {4 \, {\left (-18 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 27 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 11 i \, a^{4} + 6 \, {\left (i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
-4/3*(-18*I*a^4*e^(4*I*d*x + 4*I*c) + 27*I*a^4*e^(2*I*d*x + 2*I*c) - 11*I* a^4 + 6*(I*a^4*e^(6*I*d*x + 6*I*c) - 3*I*a^4*e^(4*I*d*x + 4*I*c) + 3*I*a^4 *e^(2*I*d*x + 2*I*c) - I*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)
Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=- \frac {8 i a^{4} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {72 i a^{4} e^{4 i c} e^{4 i d x} - 108 i a^{4} e^{2 i c} e^{2 i d x} + 44 i a^{4}}{3 d e^{6 i c} e^{6 i d x} - 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} - 3 d} \]
-8*I*a**4*log(exp(2*I*d*x) - exp(-2*I*c))/d + (72*I*a**4*exp(4*I*c)*exp(4* I*d*x) - 108*I*a**4*exp(2*I*c)*exp(2*I*d*x) + 44*I*a**4)/(3*d*exp(6*I*c)*e xp(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 = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {24 \, {\left (d x + c\right )} a^{4} + 12 i \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 i \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {21 \, a^{4} \tan \left (d x + c\right )^{2} - 6 i \, a^{4} \tan \left (d x + c\right ) - a^{4}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
1/3*(24*(d*x + c)*a^4 + 12*I*a^4*log(tan(d*x + c)^2 + 1) - 24*I*a^4*log(ta n(d*x + c)) + (21*a^4*tan(d*x + c)^2 - 6*I*a^4*tan(d*x + c) - a^4)/tan(d*x + c)^3)/d
Time = 1.50 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.42 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 384 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 192 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 87 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-352 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 87 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
1/24*(a^4*tan(1/2*d*x + 1/2*c)^3 - 12*I*a^4*tan(1/2*d*x + 1/2*c)^2 + 384*I *a^4*log(tan(1/2*d*x + 1/2*c) + I) - 192*I*a^4*log(tan(1/2*d*x + 1/2*c)) - 87*a^4*tan(1/2*d*x + 1/2*c) - (-352*I*a^4*tan(1/2*d*x + 1/2*c)^3 - 87*a^4 *tan(1/2*d*x + 1/2*c)^2 + 12*I*a^4*tan(1/2*d*x + 1/2*c) + a^4)/tan(1/2*d*x + 1/2*c)^3)/d
Time = 5.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {7\,a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {16\,a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a^4\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d}-\frac {a^4\,{\mathrm {cot}\left (c+d\,x\right )}^2\,2{}\mathrm {i}}{d} \]