Optimal. Leaf size=348 \[ \frac{\sqrt{1-c x} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 c^2 \sqrt{c x-1}}-\frac{9 \sqrt{1-c x} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 c^2 \sqrt{c x-1}}+\frac{9 \sqrt{1-c x} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}-\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.05554, antiderivative size = 429, normalized size of antiderivative = 1.23, number of steps used = 23, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5798, 5778, 5700, 3312, 3303, 3298, 3301, 5780, 5448} \[ \frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{9 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 \sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{9 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{x (c x+1)^{3/2} \sqrt{1-c^2 x^2} (1-c x)^2}{b c \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5778
Rule 5700
Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rule 5780
Rule 5448
Rubi steps
\begin{align*} \int \frac{x \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 (-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 (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} \int \frac{-1+c^2 x^2}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 c \sqrt{1-c^2 x^2}\right ) \int \frac{x^2 \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 (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} \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (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 (i \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 (a+b x)}-\frac{i \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{8 (a+b x)}-\frac{\sinh (3 x)}{16 (a+b x)}+\frac{\sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (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} \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (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 (5 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (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{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{9 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{5 a}{b}\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.913682, size = 327, normalized size = 0.94 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (-2 \sinh \left (\frac{a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+9 \sinh \left (\frac{3 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-5 a \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-5 b \sinh \left (\frac{5 a}{b}\right ) \cosh ^{-1}(c x) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 a \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+2 b \cosh \left (\frac{a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-9 a \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-9 b \cosh \left (\frac{3 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+5 a \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+5 b \cosh \left (\frac{5 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-16 b c^5 x^5+32 b c^3 x^3-16 b c x\right )}{16 b^2 c^2 \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.276, size = 1029, normalized size = 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^{5} - 2 \, c^{2} x^{3} + x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{5} x^{6} - 2 \, c^{3} x^{4} + c x^{2}\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 (5 \,{\left (c^{5} x^{5} - c^{3} x^{3}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} +{\left (10 \, c^{6} x^{6} - 17 \, c^{4} x^{4} + 8 \, c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (5 \, c^{7} x^{7} - 12 \, c^{5} x^{5} + 9 \, c^{3} x^{3} - 2 \, c x\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^{3} - x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{\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{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{{\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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