Optimal. Leaf size=198 \[ -\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.351189, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5713, 5688, 5715, 3716, 2190, 2279, 2391} \[ -\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5688
Rule 5715
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.438871, size = 126, normalized size = 0.64 \[ \frac{\frac{\sqrt{c x-1} \sqrt{c x+1} \left (-2 b^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-2 b^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )+\left (a+b \cosh ^{-1}(c x)\right ) \left (a+b \cosh ^{-1}(c x)-2 b \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.216, size = 578, normalized size = 2.9 \begin{align*}{\frac{{a}^{2}x}{d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{ab\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{ab\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}-1 \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }} \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{a b c \sqrt{-\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{d} + b^{2} \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} + \frac{2 \, a b x \operatorname{arcosh}\left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a^{2} x}{\sqrt{-c^{2} d x^{2} + d} d} \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 )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{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{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{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 (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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