Optimal. Leaf size=54 \[ -\frac {\sqrt {1+\frac {a^2}{x^2}}}{3 a^3}+\frac {\left (1+\frac {a^2}{x^2}\right )^{3/2}}{9 a^3}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5870, 6419,
272, 45} \begin {gather*} \frac {\left (\frac {a^2}{x^2}+1\right )^{3/2}}{9 a^3}-\frac {\sqrt {\frac {a^2}{x^2}+1}}{3 a^3}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 5870
Rule 6419
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx &=\int \frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x^4} \, dx\\ &=-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{3 x^3}-\frac {1}{3} a \int \frac {1}{\sqrt {1+\frac {a^2}{x^2}} x^5} \, dx\\ &=-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{3 x^3}+\frac {1}{6} a \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{3 x^3}+\frac {1}{6} a \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {\sqrt {1+\frac {a^2}{x^2}}}{3 a^3}+\frac {\left (1+\frac {a^2}{x^2}\right )^{3/2}}{9 a^3}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.89 \begin {gather*} \left (-\frac {2}{9 a^3}+\frac {1}{9 a x^2}\right ) \sqrt {\frac {a^2+x^2}{x^2}}-\frac {\sinh ^{-1}\left (\frac {a}{x}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 53, normalized size = 0.98
method | result | size |
derivativedivides | \(-\frac {\frac {a^{3} \arcsinh \left (\frac {a}{x}\right )}{3 x^{3}}-\frac {a^{2} \sqrt {1+\frac {a^{2}}{x^{2}}}}{9 x^{2}}+\frac {2 \sqrt {1+\frac {a^{2}}{x^{2}}}}{9}}{a^{3}}\) | \(53\) |
default | \(-\frac {\frac {a^{3} \arcsinh \left (\frac {a}{x}\right )}{3 x^{3}}-\frac {a^{2} \sqrt {1+\frac {a^{2}}{x^{2}}}}{9 x^{2}}+\frac {2 \sqrt {1+\frac {a^{2}}{x^{2}}}}{9}}{a^{3}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 47, normalized size = 0.87 \begin {gather*} \frac {1}{9} \, a {\left (\frac {{\left (\frac {a^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}}}{a^{4}} - \frac {3 \, \sqrt {\frac {a^{2}}{x^{2}} + 1}}{a^{4}}\right )} - \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 62, normalized size = 1.15 \begin {gather*} -\frac {3 \, a^{3} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) - {\left (a^{2} x - 2 \, x^{3}\right )} \sqrt {\frac {a^{2} + x^{2}}{x^{2}}}}{9 \, a^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 75, normalized size = 1.39 \begin {gather*} -\frac {\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right )}{3 \, x^{3}} - \frac {4 \, {\left (a^{2} - 3 \, {\left (x - \sqrt {a^{2} + x^{2}}\right )}^{2}\right )} a}{9 \, {\left (a^{2} - {\left (x - \sqrt {a^{2} + x^{2}}\right )}^{2}\right )}^{3} \mathrm {sgn}\left (x\right )} \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 = -0.02 \begin {gather*} \int \frac {\mathrm {asinh}\left (\frac {a}{x}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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