Optimal. Leaf size=169 \[ \frac {i \left (m^2+2 m+3\right ) x (e x)^m \, _2F_1\left (1,\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {e^{2 i a}}{x^2}\right )}{m+1}-\frac {i x \left (1+\frac {e^{2 i a}}{x^2}\right )^2 (e x)^m}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}-\frac {i x \left (-\frac {e^{2 i a} (1-m)}{x^2}+m+3\right ) (e x)^m}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )}+\frac {i (1-m) m x (e x)^m}{2 (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int (e x)^m \cot ^3(a+i \log (x)) \, dx &=\int (e x)^m \cot ^3(a+i \log (x)) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 122, normalized size = 0.72 \[ -\frac {i x (e x)^m \left (6 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )-12 \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )+8 \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};x^2 (\cos (2 a)-i \sin (2 a))\right )-1\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-i \, x^{6} - 3 i \, x^{4} e^{\left (2 i \, a\right )} - 3 i \, x^{2} e^{\left (4 i \, a\right )} - i \, e^{\left (6 i \, a\right )}\right )} e^{\left (m \log \relax (e) + m \log \relax (x)\right )}}{x^{6} - 3 \, x^{4} e^{\left (2 i \, a\right )} + 3 \, x^{2} e^{\left (4 i \, a\right )} - e^{\left (6 i \, a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left (a + i \, \log \relax (x)\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\cot ^{3}\left (a +i \ln \relax (x )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left (a + i \, \log \relax (x)\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (a+\ln \relax (x)\,1{}\mathrm {i}\right )}^3\,{\left (e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot ^{3}{\left (a + i \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________