Optimal. Leaf size=339 \[ \frac{15 \sqrt{1-c x} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c \sqrt{c x-1}}-\frac{15 \sqrt{1-c x} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c \sqrt{c x-1}} \]
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Rubi [A] time = 0.568075, antiderivative size = 430, normalized size of antiderivative = 1.27, number of steps used = 13, 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} \[ \frac{15 \sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 \sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c \sqrt{c x-1} \sqrt{c x+1}} \]
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{\left (1-c^2 x^2\right )^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2}}{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 ^6(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{5}{16 (a+b x)}-\frac{15 \cosh (2 x)}{32 (a+b x)}+\frac{3 \cosh (4 x)}{16 (a+b x)}-\frac{\cosh (6 x)}{32 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (15 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (15 \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 )}{32 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2} \cosh \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 )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \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 )}{32 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (15 \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 )}{32 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2} \sinh \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 )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \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 )}{32 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15 \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 )}{32 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15 \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 )}{32 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.787367, size = 191, normalized size = 0.56 \[ \frac{\sqrt{1-c^2 x^2} \left (15 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-6 \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-15 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+6 \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-10 \log \left (a+b \cosh ^{-1}(c x)\right )\right )}{32 b c \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.167, size = 591, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ( 64\,cx+64 \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 ( 1,6\,{\rm arccosh} \left (cx\right )+6\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+6\,a}{b}}}}}+{\frac{1}{ \left ( 64\,cx+64 \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,-6\,{\rm arccosh} \left (cx\right )-6\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-6\,a}{b}}}}}-{\frac{5\,\ln \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }{16\,cb}\sqrt{-{c}^{2}{x}^{2}+1}{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{3}{ \left ( 32\,cx+32 \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 ( 1,4\,{\rm arccosh} \left (cx\right )+4\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+4\,a}{b}}}}}+{\frac{15}{ \left ( 64\,cx+64 \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{15}{ \left ( 64\,cx+64 \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{3}{ \left ( 32\,cx+32 \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,-4\,{\rm arccosh} \left (cx\right )-4\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-4\,a}{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{{\left (-c^{2} x^{2} + 1\right )}^{\frac{5}{2}}}{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{{\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \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(-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{5}{2}}}{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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