Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{\left (c^2 x^2+1\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]
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Rubi [A] time = 0.14518, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx &=\int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end{align*}
Mathematica [A] time = 4.59578, size = 0, normalized size = 0. \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.44, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{b x^{3} \operatorname{arsinh}\left (c x\right ) + a x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c^{2} x^{2} + 1\right )^{\frac{3}{2}}}{x^{3} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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