Optimal. Leaf size=87 \[ \frac {(e x)^{m+1}}{e (m+1)}-\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.05, 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 = {} \[ \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 13.48, size = 158, normalized size = 1.82 \[ \frac {x (e x)^m \left (-\frac {(m+1) e^{2 a d} \left (c x^n\right )^{2 b d} \, _2F_1\left (1,\frac {m+2 b d n+1}{2 b d n};\frac {m+4 b d n+1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d n+m+1}-\, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \coth \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.58, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {e^{m} x x^{m}}{m + 1} - e^{m} \int \frac {x^{m}}{c^{b d} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + 1}\,{d x} + e^{m} \int \frac {x^{m}}{c^{b d} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________