Optimal. Leaf size=160 \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}+\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \]
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Rubi [A] time = 0.750641, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5654, 5718, 5781, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}+\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 5654
Rule 5718
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \, dx &=x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{1}{2} (5 b c) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac{1}{4} \left (15 b^2\right ) \int \sqrt{a+b \cosh ^{-1}(c x)} \, dx\\ &=\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{1}{8} \left (15 b^3 c\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx\\ &=\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c}\\ &=\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c}\\ &=\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 c}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 c}\\ &=\frac{15}{4} b^2 x \sqrt{a+b \cosh ^{-1}(c x)}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac{15 b^{5/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}-\frac{15 b^{5/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}\\ \end{align*}
Mathematica [B] time = 2.3507, size = 452, normalized size = 2.82 \[ \frac{8 a^2 e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left (\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}}+\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}}\right )-\sqrt{\pi } \sqrt{b} \left (4 a^2-12 a b+15 b^2\right ) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )-\sqrt{\pi } \sqrt{b} \left (4 a^2+12 a b+15 b^2\right ) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+4 a b \left (\frac{\sqrt{\pi } (2 a-3 b) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b}}+\frac{\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b}}-12 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{a+b \cosh ^{-1}(c x)}+8 c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right )+4 b \sqrt{a+b \cosh ^{-1}(c x)} \left (2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (a-5 b \cosh ^{-1}(c x)\right )+b c x \left (4 \cosh ^{-1}(c x)^2+15\right )\right )}{16 c} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{{\frac{5}{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{5}{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(-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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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