Optimal. Leaf size=197 \[ -\frac{\sqrt{1-c x} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b c^2 \sqrt{c x-1}} \]
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Rubi [A] time = 0.571637, antiderivative size = 245, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5798, 5781, 5448, 3303, 3298, 3301} \[ -\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5781
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{x \sqrt{-1+c x} \sqrt{1+c x}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 (a+b x)}+\frac{\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (\sqrt{1-c^2 x^2} \cosh \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 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{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \sinh \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 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{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.310069, size = 127, normalized size = 0.64 \[ \frac{\sqrt{1-c^2 x^2} \left (-\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{4 c^2 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.156, size = 361, normalized size = 1.8 \begin{align*}{\frac{1}{ \left ( 8\,cx+8 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+3\,a}{b}}}}}+{\frac{1}{ \left ( 8\,cx+8 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-3\,a}{b}}}}}-{\frac{1}{ \left ( 8\,cx+8 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-a}{b}}}}}-{\frac{1}{ \left ( 8\,cx+8 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ){{\rm e}^{{\frac{a+b{\rm arccosh} \left (cx\right )}{b}}}}} \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*} \int \frac{\sqrt{-c^{2} x^{2} + 1} x}{b \operatorname{arcosh}\left (c x\right ) + a}\,{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}{b \operatorname{arcosh}\left (c x\right ) + a}, 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 \sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname{acosh}{\left (c x \right )}}\, 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}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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