Optimal. Leaf size=86 \[ \frac {1}{3} x^3 \sqrt {\frac {1}{a^2 x^4}+1}-\frac {\sqrt {\frac {a^2+\frac {1}{x^4}}{\left (a+\frac {1}{x^2}\right )^2}} \left (a+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 a^{5/2} \sqrt {\frac {1}{a^2 x^4}+1}}+\frac {x}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6336, 8, 335, 277, 220} \[ \frac {1}{3} x^3 \sqrt {\frac {1}{a^2 x^4}+1}-\frac {\sqrt {\frac {a^2+\frac {1}{x^4}}{\left (a+\frac {1}{x^2}\right )^2}} \left (a+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 a^{5/2} \sqrt {\frac {1}{a^2 x^4}+1}}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 220
Rule 277
Rule 335
Rule 6336
Rubi steps
\begin {align*} \int e^{\text {csch}^{-1}\left (a x^2\right )} x^2 \, dx &=\frac {\int 1 \, dx}{a}+\int \sqrt {1+\frac {1}{a^2 x^4}} x^2 \, dx\\ &=\frac {x}{a}-\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x^4}{a^2}}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {x}{a}+\frac {1}{3} \sqrt {1+\frac {1}{a^2 x^4}} x^3-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 a^2}\\ &=\frac {x}{a}+\frac {1}{3} \sqrt {1+\frac {1}{a^2 x^4}} x^3-\frac {\sqrt {\frac {a^2+\frac {1}{x^4}}{\left (a+\frac {1}{x^2}\right )^2}} \left (a+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 a^{5/2} \sqrt {1+\frac {1}{a^2 x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 113, normalized size = 1.31 \[ -\frac {2 \sqrt {2} x e^{-\text {csch}^{-1}\left (a x^2\right )} \left (\frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{e^{2 \text {csch}^{-1}\left (a x^2\right )}-1}\right )^{3/2} \left (\left (1-e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )^{3/2} \left (-\, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )\right )-2 e^{2 \text {csch}^{-1}\left (a x^2\right )}+1\right )}{3 a \sqrt {a x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{2} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{a}, x\right ) \]
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 \[ \int x^{2} {\left (\sqrt {\frac {1}{a^{2} x^{4}} + 1} + \frac {1}{a x^{2}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 104, normalized size = 1.21 \[ \frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, x^{2} \left (\sqrt {i a}\, x^{5} a^{2}+2 \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \EllipticF \left (x \sqrt {i a}, i\right )+x \sqrt {i a}\right )}{3 \left (a^{2} x^{4}+1\right ) \sqrt {i a}}+\frac {x}{a} \]
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 \[ \frac {x}{a} + \frac {\frac {x \Gamma \left (\frac {1}{4}\right ) \,_2F_1\left (\begin {matrix} -\frac {1}{2},\frac {1}{4} \\ \frac {5}{4} \end {matrix} ; -a^{2} x^{4} \right )}{4 \, \Gamma \left (\frac {5}{4}\right )}}{a} \]
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 \[ \int x^2\,\left (\sqrt {\frac {1}{a^2\,x^4}+1}+\frac {1}{a\,x^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.16, size = 41, normalized size = 0.48 \[ - \frac {x^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac {1}{4}\right )} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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