Optimal. Leaf size=30 \[ \text {Int}\left (\frac {\sqrt {c^2 x^2+1}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )} \, dx &=\int \frac {\sqrt {1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end {align*}
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Mathematica [A] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^4 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1}}{b x^{4} \operatorname {arsinh}\left (c x\right ) + a x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2}+1}}{x^{4} \left (a +b \arcsinh \left (c x \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {c^2\,x^2+1}}{x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} x^{2} + 1}}{x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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