Optimal. Leaf size=33 \[ -\frac {1}{2} a \tanh ^{-1}\left (\sqrt {a^2 x^4+1}\right )-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5902, 12, 266, 63, 208} \[ -\frac {1}{2} a \tanh ^{-1}\left (\sqrt {a^2 x^4+1}\right )-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 5902
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (a x^2\right )}{x^3} \, dx &=-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2}+\frac {1}{2} \int \frac {2 a}{x \sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2}+a \int \frac {1}{x \sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2}+\frac {1}{4} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^4\right )\\ &=-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^4}\right )}{2 a}\\ &=-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2}-\frac {1}{2} a \tanh ^{-1}\left (\sqrt {1+a^2 x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ -\frac {1}{2} a \tanh ^{-1}\left (\sqrt {a^2 x^4+1}\right )-\frac {\sinh ^{-1}\left (a x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 106, normalized size = 3.21 \[ -\frac {a x^{2} \log \left (-a x^{2} + \sqrt {a^{2} x^{4} + 1} + 1\right ) - a x^{2} \log \left (-a x^{2} + \sqrt {a^{2} x^{4} + 1} - 1\right ) - x^{2} \log \left (-a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - {\left (x^{2} - 1\right )} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 58, normalized size = 1.76 \[ -\frac {1}{4} \, a {\left (\log \left (\sqrt {a^{2} x^{4} + 1} + 1\right ) - \log \left (\sqrt {a^{2} x^{4} + 1} - 1\right )\right )} - \frac {\log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.85 \[ -\frac {\arcsinh \left (a \,x^{2}\right )}{2 x^{2}}-\frac {a \arctanh \left (\frac {1}{\sqrt {a^{2} x^{4}+1}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 46, normalized size = 1.39 \[ -\frac {1}{4} \, a {\left (\log \left (\sqrt {a^{2} x^{4} + 1} + 1\right ) - \log \left (\sqrt {a^{2} x^{4} + 1} - 1\right )\right )} - \frac {\operatorname {arsinh}\left (a x^{2}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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