Optimal. Leaf size=165 \[ -\frac{2 \sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 \sqrt{c+d x-1} \sqrt{c+d x+1}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
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Rubi [A] time = 0.408877, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5864, 5656, 5775, 5658, 3308, 2180, 2205, 2204} \[ -\frac{2 \sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 \sqrt{c+d x-1} \sqrt{c+d x+1}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5864
Rule 5656
Rule 5775
Rule 5658
Rule 3308
Rule 2180
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac{2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{4 \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac{2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 \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+d x)\right )}{3 b^3 d}+\frac{2 \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+d x)\right )}{3 b^3 d}\\ &=-\frac{2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{4 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{3 b^3 d}+\frac{4 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{3 b^3 d}\\ &=-\frac{2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}\\ \end{align*}
Mathematica [A] time = 0.168889, size = 219, normalized size = 1.33 \[ \frac{e^{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \left (2 e^{\frac{2 a}{b}+\cosh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-2 \left (b e^{\cosh ^{-1}(c+d x)} \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )+e^{a/b} \left (\left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )+b \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) e^{\cosh ^{-1}(c+d x)}\right )\right )\right )}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (dx+c\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 \frac{1}{{\left (b \operatorname{arcosh}\left (d x + c\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )^{\frac{5}{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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