Optimal. Leaf size=65 \[ \frac {1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2 n};\frac {1}{2} \left (3+\frac {2}{n}\right );-a^2 x^{2 n}\right )}{2 (n+2)} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5902, 12, 364} \[ \frac {1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2 n};\frac {1}{2} \left (3+\frac {2}{n}\right );-a^2 x^{2 n}\right )}{2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 364
Rule 5902
Rubi steps
\begin {align*} \int x \sinh ^{-1}\left (a x^n\right ) \, dx &=\frac {1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac {1}{2} \int \frac {a n x^{1+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=\frac {1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac {1}{2} (a n) \int \frac {x^{1+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=\frac {1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{2+n} \, _2F_1\left (\frac {1}{2},\frac {2+n}{2 n};\frac {1}{2} \left (3+\frac {2}{n}\right );-a^2 x^{2 n}\right )}{2 (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.89 \[ \frac {x^2 \left ((n+2) \sinh ^{-1}\left (a x^n\right )-a n x^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}+\frac {1}{n};\frac {3}{2}+\frac {1}{n};-a^2 x^{2 n}\right )\right )}{2 (n+2)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int x \operatorname {arsinh}\left (a x^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int x \arcsinh \left (a \,x^{n}\right )\, dx \]
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 \[ -\frac {1}{4} \, n x^{2} - a n \int \frac {x x^{n}}{2 \, {\left (a^{3} x^{3 \, n} + a x^{n} + {\left (a^{2} x^{2 \, n} + 1\right )}^{\frac {3}{2}}\right )}}\,{d x} + \frac {1}{2} \, x^{2} \log \left (a x^{n} + \sqrt {a^{2} x^{2 \, n} + 1}\right ) + n \int \frac {x}{2 \, {\left (a^{2} x^{2 \, n} + 1\right )}}\,{d x} \]
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 \[ \int x\,\mathrm {asinh}\left (a\,x^n\right ) \,d x \]
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 \[ \int x \operatorname {asinh}{\left (a x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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