Integrand size = 24, antiderivative size = 69 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-4 a^3 x+\frac {i a^3 \log (\cos (c+d x))}{d}+\frac {3 i a^3 \log (\sin (c+d x))}{d}-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d} \]
-4*a^3*x+I*a^3*ln(cos(d*x+c))/d+3*I*a^3*ln(sin(d*x+c))/d-cot(d*x+c)*(a^3+I *a^3*tan(d*x+c))/d
Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=a^3 \left (-\frac {\cot (c+d x)}{d}+\frac {3 i \log (\tan (c+d x))}{d}-\frac {4 i \log (i+\tan (c+d x))}{d}\right ) \]
Time = 0.56 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {3042, 4036, 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.
\(\displaystyle \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^3}{\tan (c+d x)^2}dx\) |
\(\Big \downarrow \) 4036 |
\(\displaystyle -\int -\cot (c+d x) (i \tan (c+d x) a+a) \left (3 i a^2-a^2 \tan (c+d x)\right )dx-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \cot (c+d x) (i \tan (c+d x) a+a) \left (3 i a^2-a^2 \tan (c+d x)\right )dx-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(i \tan (c+d x) a+a) \left (3 i a^2-a^2 \tan (c+d x)\right )}{\tan (c+d x)}dx-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 4072 |
\(\displaystyle -i a^3 \int \tan (c+d x)dx+\int \cot (c+d x) \left (3 i a^3-4 a^3 \tan (c+d x)\right )dx-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i a^3 \int \tan (c+d x)dx+\int \frac {3 i a^3-4 a^3 \tan (c+d x)}{\tan (c+d x)}dx-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \int \frac {3 i a^3-4 a^3 \tan (c+d x)}{\tan (c+d x)}dx+\frac {i a^3 \log (\cos (c+d x))}{d}-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle 3 i a^3 \int \cot (c+d x)dx+\frac {i a^3 \log (\cos (c+d x))}{d}-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}-4 a^3 x\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 i a^3 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+\frac {i a^3 \log (\cos (c+d x))}{d}-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}-4 a^3 x\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 i a^3 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+\frac {i a^3 \log (\cos (c+d x))}{d}-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}-4 a^3 x\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {3 i a^3 \log (-\sin (c+d x))}{d}+\frac {i a^3 \log (\cos (c+d x))}{d}-\frac {\cot (c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{d}-4 a^3 x\) |
-4*a^3*x + (I*a^3*Log[Cos[c + d*x]])/d + ((3*I)*a^3*Log[-Sin[c + d*x]])/d - (Cot[c + d*x]*(a^3 + I*a^3*Tan[c + d*x]))/d
3.1.29.3.1 Defintions of rubi rules used
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]
Time = 0.46 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {a^{3} \left (3 i \ln \left (\tan \left (d x +c \right )\right )-2 i \ln \left (\sec ^{2}\left (d x +c \right )\right )-4 d x -\cot \left (d x +c \right )\right )}{d}\) | \(43\) |
derivativedivides | \(-\frac {a^{3} \left (-3 i \ln \left (\tan \left (d x +c \right )\right )+\frac {1}{\tan \left (d x +c \right )}+2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+4 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(51\) |
default | \(-\frac {a^{3} \left (-3 i \ln \left (\tan \left (d x +c \right )\right )+\frac {1}{\tan \left (d x +c \right )}+2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+4 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(51\) |
norman | \(\frac {-\frac {a^{3}}{d}-4 a^{3} x \tan \left (d x +c \right )}{\tan \left (d x +c \right )}+\frac {3 i a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(68\) |
risch | \(\frac {8 a^{3} c}{d}-\frac {2 i a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {3 i a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {i a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(75\) |
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {-2 i \, a^{3} + {\left (i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3 \, {\left (-i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \]
(-2*I*a^3 + (I*a^3*e^(2*I*d*x + 2*I*c) - I*a^3)*log(e^(2*I*d*x + 2*I*c) + 1) - 3*(-I*a^3*e^(2*I*d*x + 2*I*c) + I*a^3)*log(e^(2*I*d*x + 2*I*c) - 1))/ (d*e^(2*I*d*x + 2*I*c) - d)
Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=- \frac {2 i a^{3}}{d e^{2 i c} e^{2 i d x} - d} + \frac {a^{3} \cdot \left (3 i \log {\left (e^{2 i d x} - e^{- 2 i c} \right )} + i \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}\right )}{d} \]
-2*I*a**3/(d*exp(2*I*c)*exp(2*I*d*x) - d) + a**3*(3*I*log(exp(2*I*d*x) - e xp(-2*I*c)) + I*log(exp(2*I*d*x) + exp(-2*I*c)))/d
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {4 \, {\left (d x + c\right )} a^{3} + 2 i \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 3 i \, a^{3} \log \left (\tan \left (d x + c\right )\right ) + \frac {a^{3}}{\tan \left (d x + c\right )}}{d} \]
-(4*(d*x + c)*a^3 + 2*I*a^3*log(tan(d*x + c)^2 + 1) - 3*I*a^3*log(tan(d*x + c)) + a^3/tan(d*x + c))/d
Time = 0.71 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {-2 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 16 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 2 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - 6 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-6 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
-1/2*(-2*I*a^3*log(tan(1/2*d*x + 1/2*c) + 1) + 16*I*a^3*log(tan(1/2*d*x + 1/2*c) + I) - 2*I*a^3*log(tan(1/2*d*x + 1/2*c) - 1) - 6*I*a^3*log(tan(1/2* d*x + 1/2*c)) - a^3*tan(1/2*d*x + 1/2*c) - (-6*I*a^3*tan(1/2*d*x + 1/2*c) - a^3)/tan(1/2*d*x + 1/2*c))/d
Time = 4.64 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.55 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {a^3\,\left (\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,4{}\mathrm {i}+\mathrm {cot}\left (c+d\,x\right )-\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,3{}\mathrm {i}\right )}{d} \]