Optimal. Leaf size=354 \[ -\frac{\sqrt{1-c x} \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.13541, antiderivative size = 439, normalized size of antiderivative = 1.24, number of steps used = 20, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5798, 5778, 5780, 5448, 3303, 3298, 3301} \[ -\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{4 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{4 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{x^2 (c x+1)^{3/2} \sqrt{1-c^2 x^2} (1-c x)^2}{b c \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5778
Rule 5780
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2 \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac{\sqrt{1-c^2 x^2} \int \frac{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x)^2 (1+c x)^{3/2} \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 \left (-1+c^2 x^2\right )}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (6 c \sqrt{1-c^2 x^2}\right ) \int \frac{x^3 \left (-1+c^2 x^2\right )}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x)^2 (1+c x)^{3/2} \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 ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (6 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh ^3(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)^2 (1+c x)^{3/2} \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 \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{\left (6 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{3 \sinh (2 x)}{32 (a+b x)}+\frac{\sinh (6 x)}{32 (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)^2 (1+c x)^{3/2} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\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 )}{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (9 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x)^2 (1+c x)^{3/2} \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{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 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (9 \sqrt{1-c^2 x^2} \cosh \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 )}{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\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 )}{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \sinh \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 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (9 \sqrt{1-c^2 x^2} \sinh \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 )}{16 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 )}{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x^2 (1-c x)^2 (1+c x)^{3/2} \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{2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{2 a}{b}\right )}{16 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\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 )}{4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{6 a}{b}\right )}{16 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 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 )}{4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.0655, size = 338, normalized size = 0.95 \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \left (-\sinh \left (\frac{2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac{4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+3 a \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+3 b \sinh \left (\frac{6 a}{b}\right ) \cosh ^{-1}(c x) \text{Chi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+a \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+b \cosh \left (\frac{2 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+4 a \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+4 b \cosh \left (\frac{4 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 a \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 b \cosh \left (\frac{6 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+16 b c^6 x^6-32 b c^4 x^4+16 b c^2 x^2\right )}{16 b^2 c^3 \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.355, size = 1176, normalized size = 3.3 \begin{align*} \text{result too large to display} \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^{4} x^{6} - 2 \, c^{2} x^{4} + x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{5} x^{7} - 2 \, 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 (6 \, c^{5} x^{6} - 7 \, c^{3} x^{4} + c x^{2}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} + 2 \,{\left (6 \, c^{6} x^{7} - 11 \, c^{4} x^{5} + 6 \, c^{2} x^{3} - x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} + 3 \,{\left (2 \, c^{7} x^{8} - 5 \, c^{5} x^{6} + 4 \, c^{3} x^{4} - 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{{\left (c^{2} x^{4} - x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} 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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