Optimal. Leaf size=74 \[ \frac{(d x)^{m+1} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (m+1)}-\frac{2 b c (d x)^{m+3} \text{Hypergeometric2F1}\left (1,\frac{m+3}{4},\frac{m+7}{4},c^2 x^4\right )}{d^3 (m+1) (m+3)} \]
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Rubi [A] time = 0.0397164, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6097, 16, 364} \[ \frac{(d x)^{m+1} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (m+1)}-\frac{2 b c (d x)^{m+3} \, _2F_1\left (1,\frac{m+3}{4};\frac{m+7}{4};c^2 x^4\right )}{d^3 (m+1) (m+3)} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 16
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac{(2 b c) \int \frac{x (d x)^{1+m}}{1-c^2 x^4} \, dx}{d (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac{(2 b c) \int \frac{(d x)^{2+m}}{1-c^2 x^4} \, dx}{d^2 (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac{2 b c (d x)^{3+m} \, _2F_1\left (1,\frac{3+m}{4};\frac{7+m}{4};c^2 x^4\right )}{d^3 (1+m) (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0659015, size = 64, normalized size = 0.86 \[ -\frac{x (d x)^m \left (2 b c x^2 \text{Hypergeometric2F1}\left (1,\frac{m+3}{4},\frac{m+7}{4},c^2 x^4\right )-(m+3) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )\right )}{(m+1) (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )} \left (d x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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