Optimal. Leaf size=62 \[ \frac {x^3}{3}-\frac {2}{3} x^3 \, _2F_1\left (1,\frac {3}{2 b d n};1+\frac {3}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5659, 5657,
470, 371} \begin {gather*} \frac {x^3}{3}-\frac {2}{3} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 470
Rule 5657
Rule 5659
Rubi steps
\begin {align*} \int x^2 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(62)=124\).
time = 3.51, size = 207, normalized size = 3.34 \begin {gather*} -\frac {x^3 \left (3 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 )+(3+2 b d n) \left (\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+\, _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 )+\text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text {csch}\left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sinh (b d n \log (x))\right )\right )}{9+6 b d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.94, size = 0, normalized size = 0.00 \[\int x^{2} \coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{2} \coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^2\,\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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