Optimal. Leaf size=254 \[ \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right )+\frac{c \sqrt{c x-1} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b \sqrt{1-c x}}-\frac{c \sqrt{c x-1} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b \sqrt{1-c x}}-\frac{c \sqrt{c x-1} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b \sqrt{1-c x}}+\frac{c \sqrt{c x-1} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b \sqrt{1-c x}}-\frac{15 c \sqrt{c x-1} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt{1-c x}} \]
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Rubi [A] time = 2.09303, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{3 c^2}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{3 c^4 x^2}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{c^6 x^4}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c^4 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^6 \sqrt{1-c^2 x^2}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 c \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 c \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 (a+b x)}+\frac{\cosh (2 x)}{2 (a+b x)}+\frac{\cosh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 (a+b x)}+\frac{\cosh (2 x)}{2 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15 c \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15 c \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c \sqrt{1-c^2 x^2} \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,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c \sqrt{1-c^2 x^2} \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,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c \sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c \sqrt{1-c^2 x^2} \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,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 c \sqrt{1-c^2 x^2} \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,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c \sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c \sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c \sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{15 c \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c \sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c \sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.26887, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( -{c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{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{5}{2}}}{{\left (b \operatorname{arcosh}\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^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt{-c^{2} x^{2} + 1}}{b x^{2} \operatorname{arcosh}\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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}}}{{\left (b \operatorname{arcosh}\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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