Integrand size = 24, antiderivative size = 103 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=-8 i a^4 x-\frac {a^4 \log (\cos (c+d x))}{d}-\frac {7 a^4 \log (\sin (c+d x))}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d} \]
-8*I*a^4*x-a^4*ln(cos(d*x+c))/d-7*a^4*ln(sin(d*x+c))/d-1/2*cot(d*x+c)^2*(a ^2+I*a^2*tan(d*x+c))^2/d-3*I*cot(d*x+c)*(a^4+I*a^4*tan(d*x+c))/d
Time = 0.83 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.59 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=a^4 \left (-\frac {4 i \cot (c+d x)}{d}-\frac {\cot ^2(c+d x)}{2 d}-\frac {7 \log (\tan (c+d x))}{d}+\frac {8 \log (i+\tan (c+d x))}{d}\right ) \]
a^4*(((-4*I)*Cot[c + d*x])/d - Cot[c + d*x]^2/(2*d) - (7*Log[Tan[c + d*x]] )/d + (8*Log[I + Tan[c + d*x]])/d)
Time = 0.81 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 4036, 27, 3042, 4076, 25, 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.
| \(\displaystyle \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^4}{\tan (c+d x)^3}dx\) |
| \(\Big \downarrow \) 4036 |
| \(\displaystyle -\frac {1}{2} \int -2 \cot ^2(c+d x) (i \tan (c+d x) a+a)^2 \left (3 i a^2-a^2 \tan (c+d x)\right )dx-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \cot ^2(c+d x) (i \tan (c+d x) a+a)^2 \left (3 i a^2-a^2 \tan (c+d x)\right )dx-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(i \tan (c+d x) a+a)^2 \left (3 i a^2-a^2 \tan (c+d x)\right )}{\tan (c+d x)^2}dx-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 4076 |
| \(\displaystyle \int -\cot (c+d x) (i \tan (c+d x) a+a) \left (i \tan (c+d x) a^3+7 a^3\right )dx-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot (c+d x) (i \tan (c+d x) a+a) \left (i \tan (c+d x) a^3+7 a^3\right )dx-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {(i \tan (c+d x) a+a) \left (i \tan (c+d x) a^3+7 a^3\right )}{\tan (c+d x)}dx-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 4072 |
| \(\displaystyle a^4 \int \tan (c+d x)dx-\int \cot (c+d x) \left (8 i \tan (c+d x) a^4+7 a^4\right )dx-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \int \tan (c+d x)dx-\int \frac {8 i \tan (c+d x) a^4+7 a^4}{\tan (c+d x)}dx-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\int \frac {8 i \tan (c+d x) a^4+7 a^4}{\tan (c+d x)}dx-\frac {a^4 \log (\cos (c+d x))}{d}-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle -7 a^4 \int \cot (c+d x)dx-\frac {a^4 \log (\cos (c+d x))}{d}-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-8 i a^4 x-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -7 a^4 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a^4 \log (\cos (c+d x))}{d}-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-8 i a^4 x-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 7 a^4 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a^4 \log (\cos (c+d x))}{d}-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-8 i a^4 x-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {7 a^4 \log (-\sin (c+d x))}{d}-\frac {a^4 \log (\cos (c+d x))}{d}-\frac {3 i \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-8 i a^4 x-\frac {\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}\) |
(-8*I)*a^4*x - (a^4*Log[Cos[c + d*x]])/d - (7*a^4*Log[-Sin[c + d*x]])/d - (Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - ((3*I)*Cot[c + d*x]* (a^4 + I*a^4*Tan[c + d*x]))/d
3.1.40.3.1 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_)])^(n_), x_Symbol] :> Simp[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x] )^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] + Si mp[a/(d*(b*c + a*d)*(n + 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[ e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], 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] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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 = 0.49 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.50
| method | result | size |
| parallelrisch | \(-\frac {a^{4} \left (16 i d x +8 i \cot \left (d x +c \right )+14 \ln \left (\tan \left (d x +c \right )\right )-8 \ln \left (\sec ^{2}\left (d x +c \right )\right )+\cot ^{2}\left (d x +c \right )\right )}{2 d}\) | \(52\) |
| derivativedivides | \(\frac {a^{4} \left (-\frac {1}{2 \tan \left (d x +c \right )^{2}}-\frac {4 i}{\tan \left (d x +c \right )}-7 \ln \left (\tan \left (d x +c \right )\right )+4 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-8 i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(62\) |
| default | \(\frac {a^{4} \left (-\frac {1}{2 \tan \left (d x +c \right )^{2}}-\frac {4 i}{\tan \left (d x +c \right )}-7 \ln \left (\tan \left (d x +c \right )\right )+4 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-8 i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(62\) |
| norman | \(\frac {-\frac {a^{4}}{2 d}-\frac {4 i a^{4} \tan \left (d x +c \right )}{d}-8 i a^{4} x \left (\tan ^{2}\left (d x +c \right )\right )}{\tan \left (d x +c \right )^{2}}-\frac {7 a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {4 a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(84\) |
| risch | \(\frac {16 i a^{4} c}{d}+\frac {2 a^{4} \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-4\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(86\) |
Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.34 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {10 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \, a^{4} - {\left (a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 7 \, {\left (a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
(10*a^4*e^(2*I*d*x + 2*I*c) - 8*a^4 - (a^4*e^(4*I*d*x + 4*I*c) - 2*a^4*e^( 2*I*d*x + 2*I*c) + a^4)*log(e^(2*I*d*x + 2*I*c) + 1) - 7*(a^4*e^(4*I*d*x + 4*I*c) - 2*a^4*e^(2*I*d*x + 2*I*c) + a^4)*log(e^(2*I*d*x + 2*I*c) - 1))/( d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)
Time = 0.49 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {a^{4} \left (- 7 \log {\left (e^{2 i d x} - e^{- 2 i c} \right )} - \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}\right )}{d} + \frac {10 a^{4} e^{2 i c} e^{2 i d x} - 8 a^{4}}{d e^{4 i c} e^{4 i d x} - 2 d e^{2 i c} e^{2 i d x} + d} \]
a**4*(-7*log(exp(2*I*d*x) - exp(-2*I*c)) - log(exp(2*I*d*x) + exp(-2*I*c)) )/d + (10*a**4*exp(2*I*c)*exp(2*I*d*x) - 8*a**4)/(d*exp(4*I*c)*exp(4*I*d*x ) - 2*d*exp(2*I*c)*exp(2*I*d*x) + d)
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {16 i \, {\left (d x + c\right )} a^{4} - 8 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 14 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {8 i \, a^{4} \tan \left (d x + c\right ) + a^{4}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
-1/2*(16*I*(d*x + c)*a^4 - 8*a^4*log(tan(d*x + c)^2 + 1) + 14*a^4*log(tan( d*x + c)) + (8*I*a^4*tan(d*x + c) + a^4)/tan(d*x + c)^2)/d
Time = 1.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.46 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 128 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 8 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) + 56 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 16 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {84 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
-1/8*(a^4*tan(1/2*d*x + 1/2*c)^2 + 8*a^4*log(tan(1/2*d*x + 1/2*c) + 1) - 1 28*a^4*log(tan(1/2*d*x + 1/2*c) + I) + 8*a^4*log(tan(1/2*d*x + 1/2*c) - 1) + 56*a^4*log(tan(1/2*d*x + 1/2*c)) - 16*I*a^4*tan(1/2*d*x + 1/2*c) - (84* a^4*tan(1/2*d*x + 1/2*c)^2 - 16*I*a^4*tan(1/2*d*x + 1/2*c) - a^4)/tan(1/2* d*x + 1/2*c)^2)/d
Time = 4.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.63 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {8\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a^4\,{\mathrm {cot}\left (c+d\,x\right )}^2}{2\,d}-\frac {7\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {a^4\,\mathrm {cot}\left (c+d\,x\right )\,4{}\mathrm {i}}{d} \]