Optimal. Leaf size=102 \[ \frac{x^{m+1} \sinh ^{-1}(a x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )}{m+1}-\frac{a x^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-a^2 x^2\right )}{m^2+3 m+2} \]
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Rubi [A] time = 0.0715032, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {5762} \[ \frac{x^{m+1} \sinh ^{-1}(a x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right )}{m+1}-\frac{a x^{m+2} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;-a^2 x^2\right )}{m^2+3 m+2} \]
Antiderivative was successfully verified.
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Rule 5762
Rubi steps
\begin{align*} \int \frac{x^m \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^{1+m} \sinh ^{-1}(a x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};-a^2 x^2\right )}{1+m}-\frac{a x^{2+m} \, _3F_2\left (1,1+\frac{m}{2},1+\frac{m}{2};\frac{3}{2}+\frac{m}{2},2+\frac{m}{2};-a^2 x^2\right )}{2+3 m+m^2}\\ \end{align*}
Mathematica [A] time = 0.0312955, size = 97, normalized size = 0.95 \[ \frac{x^{m+1} \left ((m+2) \sinh ^{-1}(a x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )-a x \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-a^2 x^2\right )\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\it Arcsinh} \left ( ax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \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 (\frac{x^{m} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}, 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 \frac{x^{m} \operatorname{asinh}{\left (a x \right )}}{\sqrt{a^{2} x^{2} + 1}}\, 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 \frac{x^{m} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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