Optimal. Leaf size=56 \[ x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{n+1} \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5900, 12, 364} \[ x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{n+1} \]
Antiderivative was successfully verified.
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Rule 12
Rule 364
Rule 5900
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (a x^n\right ) \, dx &=x \sinh ^{-1}\left (a x^n\right )-\int \frac {a n x^n}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=x \sinh ^{-1}\left (a x^n\right )-(a n) \int \frac {x^n}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 1.00 \[ x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{n+1} \]
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 \[ \int \operatorname {arsinh}\left (a x^{n}\right )\,{d x} \]
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 \[ \int \arcsinh \left (a \,x^{n}\right )\, dx \]
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 \[ -a n \int \frac {x^{n}}{a^{3} x^{3 \, n} + a x^{n} + {\left (a^{2} x^{2 \, n} + 1\right )}^{\frac {3}{2}}}\,{d x} - n x + n \int \frac {1}{a^{2} x^{2 \, n} + 1}\,{d x} + x \log \left (a x^{n} + \sqrt {a^{2} x^{2 \, n} + 1}\right ) \]
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 \[ \int \mathrm {asinh}\left (a\,x^n\right ) \,d x \]
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 \[ \int \operatorname {asinh}{\left (a x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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