Optimal. Leaf size=137 \[ \frac{2 a^2 x^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},-a^2 x^2\right )}{m^3+6 m^2+11 m+6}-\frac{2 a x^{m+2} \sinh ^{-1}(a x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}+\frac{x^{m+1} \sinh ^{-1}(a x)^2}{m+1} \]
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Rubi [A] time = 0.0998867, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5661, 5762} \[ \frac{2 a^2 x^{m+3} \, _3F_2\left (1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};-a^2 x^2\right )}{m^3+6 m^2+11 m+6}-\frac{2 a x^{m+2} \sinh ^{-1}(a x) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m^2+3 m+2}+\frac{x^{m+1} \sinh ^{-1}(a x)^2}{m+1} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 5762
Rubi steps
\begin{align*} \int x^m \sinh ^{-1}(a x)^2 \, dx &=\frac{x^{1+m} \sinh ^{-1}(a x)^2}{1+m}-\frac{(2 a) \int \frac{x^{1+m} \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{1+m}\\ &=\frac{x^{1+m} \sinh ^{-1}(a x)^2}{1+m}-\frac{2 a x^{2+m} \sinh ^{-1}(a x) \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}+\frac{2 a^2 x^{3+m} \, _3F_2\left (1,\frac{3}{2}+\frac{m}{2},\frac{3}{2}+\frac{m}{2};2+\frac{m}{2},\frac{5}{2}+\frac{m}{2};-a^2 x^2\right )}{6+11 m+6 m^2+m^3}\\ \end{align*}
Mathematica [A] time = 0.0382418, size = 123, normalized size = 0.9 \[ \frac{x^{m+1} \left (2 a^2 x^2 \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},-a^2 x^2\right )+(m+3) \sinh ^{-1}(a x) \left ((m+2) \sinh ^{-1}(a x)-2 a x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.55, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\, 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 (x^{m} \operatorname{arsinh}\left (a x\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{asinh}^{2}{\left (a x \right )}\, dx \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 x^{m} \operatorname{arsinh}\left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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