Optimal. Leaf size=97 \[ -\frac{2 \text{Unintegrable}\left (\frac{c^2 x^2+1}{x^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c}+\frac{2 c \text{Unintegrable}\left (\frac{c^2 x^2+1}{x \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b}-\frac{\left (c^2 x^2+1\right )^2}{b c x^2 \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.251591, 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 = {} \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
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^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{\left (1+c^2 x^2\right )^2}{b c x^2 \left (a+b \sinh ^{-1}(c x)\right )}-\frac{2 \int \frac{1+c^2 x^2}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}+\frac{(2 c) \int \frac{1+c^2 x^2}{x \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}\\ \end{align*}
Mathematica [A] time = 4.11425, size = 0, normalized size = 0. \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.345, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}} \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*} -\frac{{\left (c^{4} x^{4} + 2 \, c^{2} x^{2} + 1\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{5} x^{5} + 2 \, c^{3} x^{3} + c x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{3} x^{4} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x^{3} + a b c x^{2} +{\left (b^{2} c^{3} x^{4} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x^{3} + b^{2} c x^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (2 \, c^{5} x^{5} - c^{3} x^{3} - 3 \, c x\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 2 \,{\left (2 \, c^{6} x^{6} + c^{4} x^{4} - 2 \, c^{2} x^{2} - 1\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (2 \, c^{7} x^{7} + 3 \, c^{5} x^{5} - c x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{5} x^{7} +{\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{5} + 2 \, a b c^{3} x^{5} + a b c x^{3} +{\left (b^{2} c^{5} x^{7} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{5} + 2 \, b^{2} c^{3} x^{5} + b^{2} c x^{3} + 2 \,{\left (b^{2} c^{4} x^{6} + b^{2} c^{2} x^{4}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{4} x^{6} + a b c^{2} x^{4}\right )} \sqrt{c^{2} x^{2} + 1}}\,{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^{2} x^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{2}}, 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^{2} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}\, 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 )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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