Optimal. Leaf size=65 \[ -\frac {a n x^{n-1} \, _2F_1\left (\frac {1}{2},-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1-n}-\frac {\sinh ^{-1}\left (a x^n\right )}{x} \]
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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5902, 12, 364} \[ -\frac {a n x^{n-1} \, _2F_1\left (\frac {1}{2},-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1-n}-\frac {\sinh ^{-1}\left (a x^n\right )}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 364
Rule 5902
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (a x^n\right )}{x^2} \, dx &=-\frac {\sinh ^{-1}\left (a x^n\right )}{x}+\int \frac {a n x^{-2+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=-\frac {\sinh ^{-1}\left (a x^n\right )}{x}+(a n) \int \frac {x^{-2+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=-\frac {\sinh ^{-1}\left (a x^n\right )}{x}-\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 = 0.06, size = 61, normalized size = 0.94 \[ \frac {a n x^{n-1} \, _2F_1\left (\frac {1}{2},\frac {n-1}{2 n};\frac {n-1}{2 n}+1;-a^2 x^{2 n}\right )}{n-1}-\frac {\sinh ^{-1}\left (a x^n\right )}{x} \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\arcsinh \left (a \,x^{n}\right )}{x^{2}}\, dx \]
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 \[ a n \int \frac {x^{n}}{a^{3} x^{2} x^{3 \, n} + a x^{2} x^{n} + {\left (a^{2} x^{2} x^{2 \, n} + x^{2}\right )} \sqrt {a^{2} x^{2 \, n} + 1}}\,{d x} - n \int \frac {1}{a^{2} x^{2} x^{2 \, n} + x^{2}}\,{d x} - \frac {n + \log \left (a x^{n} + \sqrt {a^{2} x^{2 \, n} + 1}\right )}{x} \]
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 \[ \int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x^{n} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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