Optimal. Leaf size=139 \[ \frac{\sqrt{1-c x} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{2 b c \sqrt{c x-1}} \]
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Rubi [A] time = 0.343371, antiderivative size = 178, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {5713, 5701, 3312, 3303, 3298, 3301} \[ \frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{2 b c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5701
Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{\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{\sinh ^2(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \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 )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\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 c \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{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\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 )}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\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 )}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.201807, size = 105, normalized size = 0.76 \[ \frac{\sqrt{-(c x-1) (c x+1)} \left (\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\log \left (a+b \cosh ^{-1}(c x)\right )\right )}{2 b c \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.106, size = 227, normalized size = 1.6 \begin{align*}{\frac{1}{ \left ( 4\,cx+4 \right ) \left ( cx-1 \right ) cb}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,2\,{\rm arccosh} \left (cx\right )+2\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+2\,a}{b}}}}}+{\frac{1}{ \left ( 4\,cx+4 \right ) \left ( cx-1 \right ) cb}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-2\,{\rm arccosh} \left (cx\right )-2\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-2\,a}{b}}}}}-{\frac{\ln \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }{2\,cb}\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*} \int \frac{\sqrt{-c^{2} x^{2} + 1}}{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}}{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{\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}}{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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