Optimal. Leaf size=136 \[ -\frac {2 x^3 \, _2F_1\left (1,\frac {3}{2 b d n};1+\frac {3}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac {x^3 \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {1}{3} x^3 \left (\frac {3}{b d n}+1\right ) \]
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Rubi [F] time = 0.06, 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 = {} \[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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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*}
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Mathematica [A] time = 4.62, 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.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.12, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\coth ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {b c^{2 \, b d} d n x^{3} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - {\left (b d n + 6\right )} x^{3}}{3 \, {\left (b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n\right )}} - 3 \, \int \frac {x^{2}}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n}\,{d x} + 3 \, \int \frac {x^{2}}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \coth ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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