Optimal. Leaf size=31 \[ -\frac {1}{3} a^2 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{3 a x^3} \]
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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6336, 30, 261} \[ -\frac {1}{3} a^2 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 261
Rule 6336
Rubi steps
\begin {align*} \int \frac {e^{\text {csch}^{-1}(a x)}}{x^3} \, dx &=\frac {\int \frac {1}{x^4} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x^3} \, dx\\ &=-\frac {1}{3} a^2 \left (1+\frac {1}{a^2 x^2}\right )^{3/2}-\frac {1}{3 a x^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 37, normalized size = 1.19 \[ -\frac {a x \sqrt {\frac {1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+1}{3 a x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 47, normalized size = 1.52 \[ -\frac {a^{3} x^{3} + {\left (a^{3} x^{3} + a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 69, normalized size = 2.23 \[ \frac {2 \, {\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a^{2} \mathrm {sgn}\relax (x) + a^{2} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 42, normalized size = 1.35 \[ -\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} x^{2}+1\right )}{3 x^{2}}-\frac {1}{3 x^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 25, normalized size = 0.81 \[ -\frac {1}{3} \, a^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - \frac {1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.16, size = 42, normalized size = 1.35 \[ -\frac {\frac {x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3}+\frac {1}{3\,a}}{x^3}-\frac {a^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3} \]
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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