Optimal. Leaf size=197 \[ -\frac {2 a \sqrt {a^2 x^4+1}}{3 x}+\frac {2 a^2 x \sqrt {a^2 x^4+1}}{3 \left (a x^2+1\right )}+\frac {a^{3/2} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {a^2 x^4+1}}-\frac {2 a^{3/2} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {a^2 x^4+1}}-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5902, 12, 325, 305, 220, 1196} \[ \frac {2 a^2 x \sqrt {a^2 x^4+1}}{3 \left (a x^2+1\right )}-\frac {2 a \sqrt {a^2 x^4+1}}{3 x}+\frac {a^{3/2} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {a^2 x^4+1}}-\frac {2 a^{3/2} \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {a^2 x^4+1}}-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 220
Rule 305
Rule 325
Rule 1196
Rule 5902
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (a x^2\right )}{x^4} \, dx &=-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3}+\frac {1}{3} \int \frac {2 a}{x^2 \sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3}+\frac {1}{3} (2 a) \int \frac {1}{x^2 \sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {2 a \sqrt {1+a^2 x^4}}{3 x}-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \int \frac {x^2}{\sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {2 a \sqrt {1+a^2 x^4}}{3 x}-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3}+\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {1-a x^2}{\sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {2 a \sqrt {1+a^2 x^4}}{3 x}+\frac {2 a^2 x \sqrt {1+a^2 x^4}}{3 \left (1+a x^2\right )}-\frac {\sinh ^{-1}\left (a x^2\right )}{3 x^3}-\frac {2 a^{3/2} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {1+a^2 x^4}}+\frac {a^{3/2} \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 \sqrt {1+a^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 88, normalized size = 0.45 \[ \frac {1}{3} \left (-\frac {2 a \sqrt {a^2 x^4+1}}{x}+\frac {2 a^2 \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {i a} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {i a} x\right )\right |-1\right )\right )}{\sqrt {i a}}-\frac {\sinh ^{-1}\left (a x^2\right )}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 101, normalized size = 0.51 \[ -\frac {\arcsinh \left (a \,x^{2}\right )}{3 x^{3}}+\frac {2 a \left (-\frac {\sqrt {a^{2} x^{4}+1}}{x}+\frac {i a \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {i a}, i\right )-\EllipticE \left (x \sqrt {i a}, i\right )\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3} \]
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 \[ -\frac {1}{12} i \, \sqrt {2} a^{\frac {3}{2}} {\left (\log \left (\frac {i \, \sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right ) - \log \left (-\frac {i \, \sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right )\right )} - \frac {1}{12} i \, \sqrt {2} a^{\frac {3}{2}} {\left (\log \left (\frac {i \, \sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right ) - \log \left (-\frac {i \, \sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right )\right )} + \frac {1}{12} \, \sqrt {2} a^{\frac {3}{2}} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right ) - \frac {1}{12} \, \sqrt {2} a^{\frac {3}{2}} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right ) + 2 \, a \int \frac {1}{3 \, {\left (a^{3} x^{8} + a x^{4} + {\left (a^{2} x^{6} + x^{2}\right )} \sqrt {a^{2} x^{4} + 1}\right )}}\,{d x} - \frac {\log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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