Integrand size = 17, antiderivative size = 63 \[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^3}{3}-\frac {2}{3} x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5658, 5656, 470, 371} \[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^3}{3}-\frac {2}{3} x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
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Rule 371
Rule 470
Rule 5656
Rule 5658
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \tanh (d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{n}} \left (-1+e^{2 a d} x^{2 b d}\right )}{1+e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {x^3}{3}-\frac {\left (2 x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{n}}}{1+e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {x^3}{3}-\frac {2}{3} x^3 \operatorname {Hypergeometric2F1}\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{align*}
Leaf count is larger than twice the leaf count of optimal. \(136\) vs. \(2(63)=126\).
Time = 7.82 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.16 \[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^3 \left (3 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1+\frac {3}{2 b d n},2+\frac {3}{2 b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(3+2 b d n) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{9+6 b d n} \]
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\[\int x^{2} \tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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\[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int x^2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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