Optimal. Leaf size=169 \[ -\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d e n}+\frac {(e x)^{m+1} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d e n \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}+\frac {(e x)^{m+1} (b d n+m+1)}{b d e (m+1) n} \]
[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 = {} \[ \int (e x)^m \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int (e x)^m \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 16.80, size = 516, normalized size = 3.05 \[ \frac {(m+1) x^{-m} (e x)^m \text {sech}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac {x^{m+1} \sinh (b d n \log (x)) \text {sech}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac {\cosh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac {(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 b d n+m+1) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(m+1) \exp \left (\frac {a (2 b d n+2 m+1)}{b n}+\frac {(2 b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 b d n+m+1)\right ) \, _2F_1\left (1,\frac {m+2 b d n+1}{2 b d n};\frac {m+4 b d n+1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+m+1) \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 b d n+m+1)}\right )}{b d n}-\frac {x (e x)^m \sinh (b d n \log (x)) \text {sech}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \text {sech}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{b d n}+\frac {x (e x)^m}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.48, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\tanh ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, e^{m} {\left (m + 1\right )} \int \frac {x^{m}}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n}\,{d x} + \frac {b c^{2 \, b d} d e^{m} n x e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d + m \log \relax (x)\right )} + {\left (b d e^{m} n + 2 \, e^{m} {\left (m + 1\right )}\right )} x x^{m}}{{\left (m n + n\right )} b c^{2 \, b d} d e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + {\left (m n + n\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2\,{\left (e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \tanh ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________