Optimal. Leaf size=105 \[ \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )},x\right )+\frac{c \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b}-\frac{c \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b}+\frac{3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{2 b} \]
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Rubi [A] time = 0.604161, 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 = {} \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]
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 )} \, dx &=\int \left (\frac{2 c^2}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}+\frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}+\frac{c^4 x^2}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}\right ) \, dx\\ &=\left (2 c^2\right ) \int \frac{1}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+c^4 \int \frac{x^2}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac{2 c \log \left (a+b \sinh ^{-1}(c x)\right )}{b}+c \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac{2 c \log \left (a+b \sinh ^{-1}(c x)\right )}{b}-c \operatorname{Subst}\left (\int \left (\frac{1}{2 (a+b x)}-\frac{\cosh (2 x)}{2 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac{3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{2 b}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac{3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{2 b}+\frac{1}{2} \left (c \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )-\frac{1}{2} \left (c \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac{c \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b}+\frac{3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{2 b}-\frac{c \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b}+\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end{align*}
Mathematica [A] time = 1.3949, size = 0, normalized size = 0. \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.143, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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^{2}}\,{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^{2} \operatorname{arsinh}\left (c x\right ) + a 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 )}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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