Optimal. Leaf size=31 \[ -\frac {1}{6} a^2 \left (\frac {1}{a^2 x^4}+1\right )^{3/2}-\frac {1}{6 a x^6} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6336, 30, 261} \[ -\frac {1}{6} a^2 \left (\frac {1}{a^2 x^4}+1\right )^{3/2}-\frac {1}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 30
Rule 261
Rule 6336
Rubi steps
\begin {align*} \int \frac {e^{\text {csch}^{-1}\left (a x^2\right )}}{x^5} \, dx &=\frac {\int \frac {1}{x^7} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^4}}}{x^5} \, dx\\ &=-\frac {1}{6} a^2 \left (1+\frac {1}{a^2 x^4}\right )^{3/2}-\frac {1}{6 a x^6}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 1.26 \[ -\frac {a x^2 \sqrt {\frac {1}{a^2 x^4}+1} \left (a^2 x^4+1\right )+1}{6 a x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 49, normalized size = 1.58 \[ -\frac {a^{3} x^{6} + {\left (a^{3} x^{6} + a x^{2}\right )} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{6 \, a x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 71, normalized size = 2.29 \[ \frac {\frac {2 \, {\left (3 \, {\left (x^{2} {\left | a \right |} - \sqrt {a^{2} x^{4} + 1}\right )}^{4} a^{4} + a^{4}\right )}}{{\left ({\left (x^{2} {\left | a \right |} - \sqrt {a^{2} x^{4} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {a}{x^{6}}}{6 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 42, normalized size = 1.35 \[ -\frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, \left (a^{2} x^{4}+1\right )}{6 x^{4}}-\frac {1}{6 x^{6} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 0.81 \[ -\frac {1}{6} \, a^{2} {\left (\frac {1}{a^{2} x^{4}} + 1\right )}^{\frac {3}{2}} - \frac {1}{6 \, a x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.16, size = 44, normalized size = 1.42 \[ -\frac {\frac {1}{6\,a}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^4}+1}}{6}}{x^6}-\frac {a^2\,\sqrt {\frac {1}{a^2\,x^4}+1}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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