Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^3},x\right ) \]
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Rubi [A] time = 0.10, 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 = {} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx &=\int \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx\\ \end {align*}
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Mathematica [A] time = 6.34, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{3}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (\frac {e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{\sqrt {d}} - \frac {\sqrt {e x^{2} + d} e}{d} + \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{2}}\right )} a + b \int \frac {\sqrt {e x^{2} + d} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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