Optimal. Leaf size=292 \[ -\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c} \]
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Rubi [A] time = 1.25055, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5664, 5759, 5718, 5658, 3308, 2180, 2205, 2204, 5670, 5448} \[ -\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c} \]
Antiderivative was successfully verified.
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Rule 5664
Rule 5759
Rule 5718
Rule 5658
Rule 3308
Rule 2180
Rule 2205
Rule 2204
Rule 5670
Rule 5448
Rubi steps
\begin{align*} \int x^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \, dx &=\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{1}{2} (b c) \int \frac{x^3 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac{1}{12} b^2 \int \frac{x^2}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx-\frac{b \int \frac{x \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}+\frac{b^2 \int \frac{1}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{6 c^2}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{b \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{6 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{\sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{b \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{12 c^3}+\frac{b \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{12 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{48 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{48 c^3}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{b \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{6 c^3}+\frac{b \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{6 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{b^{3/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{b^{3/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}-\frac{b \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{48 c^3}-\frac{b \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{48 c^3}+\frac{b \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{48 c^3}+\frac{b \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{48 c^3}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{3 b^{3/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}-\frac{b^{3/2} e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}+\frac{3 b^{3/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{b^{3/2} e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}\\ \end{align*}
Mathematica [A] time = 2.25499, size = 540, normalized size = 1.85 \[ \frac{a e^{-\frac{3 a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left (9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt{-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}}+\frac{\sqrt{b} \left (9 \left (\sqrt{\pi } (2 a-3 b) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )-12 \sqrt{b} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{a+b \cosh ^{-1}(c x)}+8 \sqrt{b} c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right )+\sqrt{3 \pi } (2 a-b) \left (\sinh \left (\frac{3 a}{b}\right )+\cosh \left (\frac{3 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+\sqrt{3 \pi } (2 a+b) \left (\cosh \left (\frac{3 a}{b}\right )-\sinh \left (\frac{3 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+12 \sqrt{b} \left (2 \cosh ^{-1}(c x) \cosh \left (3 \cosh ^{-1}(c x)\right )-\sinh \left (3 \cosh ^{-1}(c x)\right )\right ) \sqrt{a+b \cosh ^{-1}(c x)}\right )}{288 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{{\frac{3}{2}}}\, dx \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{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{3}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 x^{2} \left (a + b \operatorname{acosh}{\left (c x \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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