Optimal. Leaf size=86 \[ \frac {x}{a}+\frac {1}{3} \sqrt {1+\frac {1}{a^2 x^4}} x^3-\frac {\sqrt {\frac {a^2+\frac {1}{x^4}}{\left (a+\frac {1}{x^2}\right )^2}} \left (a+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{3 a^{5/2} \sqrt {1+\frac {1}{a^2 x^4}}} \]
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Rubi [A]
time = 0.03, 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 = {6471, 8, 342,
283, 226} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 226
Rule 283
Rule 342
Rule 6471
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}-\text {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 \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.16, size = 113, normalized size = 1.31 \begin {gather*} -\frac {2 \sqrt {2} e^{-\text {csch}^{-1}\left (a x^2\right )} \left (\frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{-1+e^{2 \text {csch}^{-1}\left (a x^2\right )}}\right )^{3/2} x \left (1-2 e^{2 \text {csch}^{-1}\left (a x^2\right )}-\left (1-e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )^{3/2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};e^{2 \text {csch}^{-1}\left (a x^2\right )}\right )\right )}{3 a \sqrt {a x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 104, normalized size = 1.21
method | result | size |
default | \(\frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, x^{2} \left (\sqrt {i a}\, a^{2} x^{5}+2 \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \EllipticF \left (x \sqrt {i a}, i\right )+x \sqrt {i a}\right )}{3 \left (a^{2} x^{4}+1\right ) \sqrt {i a}}+\frac {x}{a}\) | \(104\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.11, size = 55, normalized size = 0.64 \begin {gather*} \frac {a x^{3} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 2 \, a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}}}{x}, -1\right ) + 3 \, x}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.26, size = 41, normalized size = 0.48 \begin {gather*} - \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,\left (\sqrt {\frac {1}{a^2\,x^4}+1}+\frac {1}{a\,x^2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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