Optimal. Leaf size=154 \[ -\frac{\sqrt{1-c x} \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3 \sqrt{c x-1}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.878566, antiderivative size = 185, normalized size of antiderivative = 1.2, number of steps used = 17, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5798, 5778, 5670, 5448, 12, 3303, 3298, 3301} \[ -\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{x^2 (1-c x) \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5778
Rule 5670
Rule 5448
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{x^2 \sqrt{-1+c x} \sqrt{1+c x}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{\left (2 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 c \sqrt{1-c^2 x^2}\right ) \int \frac{x^3}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{\left (2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{\left (2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 (a+b x)}+\frac{\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \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 )}{2 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \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 )}{2 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{\sqrt{1-c^2 x^2} \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{4 a}{b}\right )}{2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.4689, size = 130, normalized size = 0.84 \[ \frac{\sqrt{1-c^2 x^2} \left (-\sinh \left (\frac{4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac{4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-2 b c^2 x^2 \left (c^2 x^2-1\right )\right )}{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.235, size = 422, normalized size = 2.7 \begin{align*}{\frac{1}{ \left ( 16\,cx+16 \right ) \left ( cx-1 \right ){c}^{3} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -8\,\sqrt{cx+1}\sqrt{cx-1}{x}^{4}{c}^{4}+8\,{c}^{5}{x}^{5}+8\,\sqrt{cx+1}\sqrt{cx-1}{x}^{2}{c}^{2}-12\,{c}^{3}{x}^{3}-\sqrt{cx-1}\sqrt{cx+1}+4\,cx \right ) }-{\frac{1}{ \left ( 4\,cx+4 \right ) \left ( cx-1 \right ){c}^{3}{b}^{2}}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,4\,{\rm arccosh} \left (cx\right )+4\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+4\,a}{b}}}}}-{\frac{1}{16\,{c}^{3}{b}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1} \left ( 8\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}b{c}^{3}+8\,{x}^{4}b{c}^{4}-4\,\sqrt{cx-1}\sqrt{cx+1}xbc-8\,{x}^{2}b{c}^{2}+4\,{\rm arccosh} \left (cx\right ){{\rm e}^{-4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-4\,{\rm arccosh} \left (cx\right )-4\,{\frac{a}{b}} \right ) b+4\,{{\rm e}^{-4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-4\,{\rm arccosh} \left (cx\right )-4\,{\frac{a}{b}} \right ) a+b \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{1}{8\,{c}^{3} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) b}\sqrt{-{c}^{2}{x}^{2}+1}{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left ({\left (c^{2} x^{4} - x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{3} x^{5} - c x^{3}\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x - a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{{\left ({\left (4 \, c^{3} x^{4} - c x^{2}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} + 2 \,{\left (4 \, c^{4} x^{5} - 4 \, c^{2} x^{3} + x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (4 \, c^{5} x^{6} - 7 \, c^{3} x^{4} + 3 \, c x^{2}\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \,{\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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