Optimal. Leaf size=136 \[ -\frac{2 x^3 \, _2F_1\left (1,\frac{3}{2 b d n};1+\frac{3}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac{x^3 \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac{1}{3} x^3 \left (\frac{3}{b d n}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.0584018, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [A] time = 4.53231, size = 165, normalized size = 1.21 \[ \frac{x^3 \left ((2 b d n+3) \left (-3 \, _2F_1\left (1,\frac{3}{2 b d n};1+\frac{3}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-3 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-9 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{3}{2 b d n};2+\frac{3}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{3 b d n (2 b d n+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b c^{2 \, b d} d n x^{3} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} -{\left (b d n + 6\right )} x^{3}}{3 \,{\left (b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n\right )}} - 3 \, \int \frac{x^{2}}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n}\,{d x} + 3 \, \int \frac{x^{2}}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]