Integrand size = 24, antiderivative size = 71 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=-8 a^4 x+\frac {4 i a^4 \log (\cos (c+d x))}{d}+\frac {4 i a^4 \log (\sin (c+d x))}{d}-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \] Output:
-8*a^4*x+4*I*a^4*ln(cos(d*x+c))/d+4*I*a^4*ln(sin(d*x+c))/d-cot(d*x+c)*(a^2 +I*a^2*tan(d*x+c))^2/d
Time = 0.59 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=a^4 \left (-\frac {\cot (c+d x)}{d}+\frac {4 i \log (\tan (c+d x))}{d}-\frac {8 i \log (i+\tan (c+d x))}{d}+\frac {\tan (c+d x)}{d}\right ) \] Input:
Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]
Output:
a^4*(-(Cot[c + d*x]/d) + ((4*I)*Log[Tan[c + d*x]])/d - ((8*I)*Log[I + Tan[ c + d*x]])/d + Tan[c + d*x]/d)
Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4036, 27, 3042, 4024, 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.
\(\displaystyle \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^4}{\tan (c+d x)^2}dx\) |
\(\Big \downarrow \) 4036 |
\(\displaystyle -\int -4 i \cot (c+d x) \left (i \tan (c+d x) a^2+a^2\right )^2dx-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 i \int \cot (c+d x) \left (i \tan (c+d x) a^2+a^2\right )^2dx-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 i \int \frac {\left (i \tan (c+d x) a^2+a^2\right )^2}{\tan (c+d x)}dx-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
\(\Big \downarrow \) 4024 |
\(\displaystyle 4 i \left (-a^4 \int \tan (c+d x)dx+a^4 \int \cot (c+d x)dx+2 i a^4 x\right )-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 i \left (-a^4 \int \tan (c+d x)dx+a^4 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+2 i a^4 x\right )-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 i \left (-a^4 \int \tan (c+d x)dx-a^4 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+2 i a^4 x\right )-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle 4 i \left (\frac {a^4 \log (-\sin (c+d x))}{d}+\frac {a^4 \log (\cos (c+d x))}{d}+2 i a^4 x\right )-\frac {\cot (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\) |
Input:
Int[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]
Output:
(4*I)*((2*I)*a^4*x + (a^4*Log[Cos[c + d*x]])/d + (a^4*Log[-Sin[c + d*x]])/ d) - (Cot[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^2)/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_)])^2/((a_.) + (b_.)*tan[(e_.) + (f _.)*(x_)]), x_Symbol] :> Simp[d*(2*b*c - a*d)*(x/b^2), x] + (Simp[d^2/b I nt[Tan[e + f*x], x], x] + Simp[(b*c - a*d)^2/b^2 Int[1/(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]
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])
Time = 1.52 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {a^{4} \left (-\cot \left (d x +c \right )+4 i \ln \left (\tan \left (d x +c \right )\right )-4 i \ln \left (\sec \left (d x +c \right )^{2}\right )-8 d x +\tan \left (d x +c \right )\right )}{d}\) | \(49\) |
risch | \(\frac {16 a^{4} c}{d}-\frac {4 i a^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {4 i a^{4} \ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-1\right )}{d}\) | \(67\) |
derivativedivides | \(\frac {a^{4} \left (\tan \left (d x +c \right )-d x -c \right )+4 i a^{4} \ln \left (\cos \left (d x +c \right )\right )-6 a^{4} \left (d x +c \right )+4 i a^{4} \ln \left (\sin \left (d x +c \right )\right )+a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(80\) |
default | \(\frac {a^{4} \left (\tan \left (d x +c \right )-d x -c \right )+4 i a^{4} \ln \left (\cos \left (d x +c \right )\right )-6 a^{4} \left (d x +c \right )+4 i a^{4} \ln \left (\sin \left (d x +c \right )\right )+a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(80\) |
norman | \(\frac {\frac {a^{4} \tan \left (d x +c \right )^{2}}{d}-\frac {a^{4}}{d}-8 a^{4} x \tan \left (d x +c \right )}{\tan \left (d x +c \right )}+\frac {4 i a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {4 i a^{4} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(83\) |
Input:
int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
a^4*(-cot(d*x+c)+4*I*ln(tan(d*x+c))-4*I*ln(sec(d*x+c)^2)-8*d*x+tan(d*x+c)) /d
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {4 \, {\left (i \, a^{4} + {\left (-i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a^{4}\right )} \log \left (e^{\left (4 i \, d x + 4 i \, c\right )} - 1\right )\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} - d} \] Input:
integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
Output:
-4*(I*a^4 + (-I*a^4*e^(4*I*d*x + 4*I*c) + I*a^4)*log(e^(4*I*d*x + 4*I*c) - 1))/(d*e^(4*I*d*x + 4*I*c) - d)
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=- \frac {4 i a^{4}}{d e^{4 i c} e^{4 i d x} - d} + \frac {4 i a^{4} \log {\left (e^{4 i d x} - e^{- 4 i c} \right )}}{d} \] Input:
integrate(cot(d*x+c)**2*(a+I*a*tan(d*x+c))**4,x)
Output:
-4*I*a**4/(d*exp(4*I*c)*exp(4*I*d*x) - d) + 4*I*a**4*log(exp(4*I*d*x) - ex p(-4*I*c))/d
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {8 \, {\left (d x + c\right )} a^{4} + 4 i \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 4 i \, a^{4} \log \left (\tan \left (d x + c\right )\right ) - a^{4} \tan \left (d x + c\right ) + \frac {a^{4}}{\tan \left (d x + c\right )}}{d} \] Input:
integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")
Output:
-(8*(d*x + c)*a^4 + 4*I*a^4*log(tan(d*x + c)^2 + 1) - 4*I*a^4*log(tan(d*x + c)) - a^4*tan(d*x + c) + a^4/tan(d*x + c))/d
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {8 i \, a^{4} \log \left (\tan \left (d x + c\right ) + i\right )}{d} + \frac {4 i \, a^{4} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} + \frac {a^{4} \tan \left (d x + c\right )}{d} - \frac {a^{4}}{d \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")
Output:
-8*I*a^4*log(tan(d*x + c) + I)/d + 4*I*a^4*log(abs(tan(d*x + c)))/d + a^4* tan(d*x + c)/d - a^4/(d*tan(d*x + c))
Time = 0.93 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {a^4\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}-\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}}{d}+\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,4{}\mathrm {i}}{d} \] Input:
int(cot(c + d*x)^2*(a + a*tan(c + d*x)*1i)^4,x)
Output:
(a^4*tan(c + d*x))/d - (a^4*cot(c + d*x))/d - (a^4*log(tan(c + d*x) + 1i)* 8i)/d + (a^4*log(tan(c + d*x))*4i)/d
Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.24 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {a^{4} \left (-8 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) i +4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right ) i +4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right ) i +4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) i -8 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d x +2 \sin \left (d x +c \right )^{2}-1\right )}{\cos \left (d x +c \right ) \sin \left (d x +c \right ) d} \] Input:
int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x)
Output:
(a**4*( - 8*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*i + 4*c os(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)*i + 4*cos(c + d*x)*log( tan((c + d*x)/2) + 1)*sin(c + d*x)*i + 4*cos(c + d*x)*log(tan((c + d*x)/2) )*sin(c + d*x)*i - 8*cos(c + d*x)*sin(c + d*x)*d*x + 2*sin(c + d*x)**2 - 1 ))/(cos(c + d*x)*sin(c + d*x)*d)