Optimal. Leaf size=413 \[ \frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 d^2}+\frac{4 a b x \sqrt{c x-1} \sqrt{c x+1}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 (1-c x) (c x+1)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.921047, antiderivative size = 424, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {5798, 5752, 5718, 5654, 74, 5766, 5694, 4182, 2279, 2391} \[ \frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 a b x \sqrt{c x-1} \sqrt{c x+1}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 (1-c x) (c x+1)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5752
Rule 5718
Rule 5654
Rule 74
Rule 5766
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \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{x^3 \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^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 (1-c x) (1+c x)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 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) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{c^3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 (1-c x) (1+c x)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^4 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^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 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^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 (1-c x) (1+c x)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^4 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 \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^4 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 \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 (1-c x) (1+c x)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^4 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.58449, size = 302, normalized size = 0.73 \[ \frac{b^2 \left (-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )-\cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2+3 \cosh ^{-1}(c x)^2-2 \cosh \left (2 \cosh ^{-1}(c x)\right )-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )+2\right )-2 a^2 \left (c^2 x^2-2\right )+2 a b \left (-\cosh ^{-1}(c x) \left (\cosh \left (2 \cosh ^{-1}(c x)\right )-3\right )+\sinh \left (2 \cosh ^{-1}(c x)\right )-2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{2 c^4 d \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.394, size = 836, normalized size = 2. \begin{align*} \text{result too large to display} \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{{\left (b^{2} x^{3} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arcosh}\left (c x\right ) + a^{2} x^{3}\right )} \sqrt{-c^{2} d x^{2} + d}}{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{x^{3} \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} x^{3}}{{\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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