Optimal. Leaf size=52 \[ \frac {1}{4} x^4 \sqrt {\frac {1}{a^2 x^4}+1}+\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^4}+1}\right )}{4 a^2}+\frac {x^2}{2 a} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6336, 30, 266, 47, 63, 208} \[ \frac {1}{4} x^4 \sqrt {\frac {1}{a^2 x^4}+1}+\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^4}+1}\right )}{4 a^2}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 47
Rule 63
Rule 208
Rule 266
Rule 6336
Rubi steps
\begin {align*} \int e^{\text {csch}^{-1}\left (a x^2\right )} x^3 \, dx &=\frac {\int x \, dx}{a}+\int \sqrt {1+\frac {1}{a^2 x^4}} x^3 \, dx\\ &=\frac {x^2}{2 a}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^4}\right )\\ &=\frac {x^2}{2 a}+\frac {1}{4} \sqrt {1+\frac {1}{a^2 x^4}} x^4-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^4}\right )}{8 a^2}\\ &=\frac {x^2}{2 a}+\frac {1}{4} \sqrt {1+\frac {1}{a^2 x^4}} x^4-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^4}}\right )\\ &=\frac {x^2}{2 a}+\frac {1}{4} \sqrt {1+\frac {1}{a^2 x^4}} x^4+\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^4}}\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 1.02 \[ \frac {a x^2 \left (a x^2 \sqrt {\frac {1}{a^2 x^4}+1}+2\right )+\log \left (x^2 \left (\sqrt {\frac {1}{a^2 x^4}+1}+1\right )\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 70, normalized size = 1.35 \[ \frac {a^{2} x^{4} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 2 \, a x^{2} - \log \left (a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} - a x^{2}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 57, normalized size = 1.10 \[ \frac {2 \, a x^{2} + {\left (\sqrt {a^{2} x^{4} + 1} x^{2} - \frac {\log \left (-x^{2} {\left | a \right |} + \sqrt {a^{2} x^{4} + 1}\right )}{{\left | a \right |}}\right )} {\left | a \right |}}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 94, normalized size = 1.81 \[ \frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, x^{2} \left (x^{2} \sqrt {\frac {a^{2} x^{4}+1}{a^{2}}}\, a^{2}+\ln \left (x^{2}+\sqrt {\frac {a^{2} x^{4}+1}{a^{2}}}\right )\right )}{4 \sqrt {\frac {a^{2} x^{4}+1}{a^{2}}}\, a^{2}}+\frac {x^{2}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 81, normalized size = 1.56 \[ \frac {x^{2}}{2 \, a} + \frac {\sqrt {\frac {1}{a^{2} x^{4}} + 1}}{4 \, {\left (a^{2} {\left (\frac {1}{a^{2} x^{4}} + 1\right )} - a^{2}\right )}} + \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{4}} + 1} + 1\right )}{8 \, a^{2}} - \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{4}} + 1} - 1\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.61, size = 42, normalized size = 0.81 \[ \frac {\mathrm {atanh}\left (\sqrt {\frac {1}{a^2\,x^4}+1}\right )}{4\,a^2}+\frac {x^4\,\sqrt {\frac {1}{a^2\,x^4}+1}}{4}+\frac {x^2}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.68, size = 36, normalized size = 0.69 \[ \frac {x^{2} \sqrt {a^{2} x^{4} + 1}}{4 a} + \frac {x^{2}}{2 a} + \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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