Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]
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Rubi [A] time = 0.5394, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
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^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=-\frac{\sqrt{1-c^2 x^2} \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.877244, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.382, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b{\rm arccosh} \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{arcosh}\left (c x\right ) + a\right )} x^{4}}\,{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^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}}, 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 (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{x^{4} \left (a + b \operatorname{acosh}{\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{arcosh}\left (c x\right ) + a\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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