Optimal. Leaf size=234 \[ -\frac{b^2 c \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{2 b c \sqrt{d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.631433, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {5798, 5738, 5660, 3718, 2190, 2279, 2391, 5676} \[ \frac{b^2 c \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{2 b c \sqrt{d-c^2 d x^2} \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]
Warning: Unable to verify antiderivative.
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Rule 5798
Rule 5738
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 5676
Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^2} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{\left (2 b c \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{3 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{b^2 c \sqrt{d-c^2 d x^2} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.62442, size = 270, normalized size = 1.15 \[ \frac{1}{3} b^2 c \sqrt{d-c^2 d x^2} \left (\frac{3 \sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{1-c x}+\cosh ^{-1}(c x) \left (\frac{\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+3\right )+6 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{3 \cosh ^{-1}(c x)}{c x}\right )\right )-\frac{a^2 \sqrt{d-c^2 d x^2}}{x}+a^2 c \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+a b c \sqrt{d-c^2 d x^2} \left (\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.336, size = 582, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{dx} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{a}^{2}{c}^{2}x\sqrt{-{c}^{2}d{x}^{2}+d}-{{a}^{2}{c}^{2}d\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{3}c}{3}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}c\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}x{c}^{2}}{ \left ( cx+1 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{ \left ( cx+1 \right ) \left ( cx-1 \right ) x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) c}{\sqrt{cx-1}\sqrt{cx+1}}}+{{b}^{2}c\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{ab \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}c\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )c}{\sqrt{cx-1}\sqrt{cx+1}}}-2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )x{c}^{2}}{ \left ( cx+1 \right ) \left ( cx-1 \right ) }}+2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )}{ \left ( cx+1 \right ) \left ( cx-1 \right ) x}}+2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) c}{\sqrt{cx-1}\sqrt{cx+1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{x^{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{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{x^{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{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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