Optimal. Leaf size=140 \[ -\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.401951, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5654, 5718, 5658, 3308, 2180, 2205, 2204} \[ -\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5654
Rule 5718
Rule 5658
Rule 3308
Rule 2180
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}(c x)\right )^{3/2} \, dx &=x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{1}{2} (3 b c) \int \frac{x \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{3 b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}+\frac{1}{4} \left (3 b^2\right ) \int \frac{1}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx\\ &=-\frac{3 b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{(3 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 )}{4 c}\\ &=-\frac{3 b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{(3 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 )}{8 c}+\frac{(3 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 )}{8 c}\\ &=-\frac{3 b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}-\frac{(3 b) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{4 c}+\frac{(3 b) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{4 c}\\ &=-\frac{3 b \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \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 )}{8 c}+\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 )}{8 c}\\ \end{align*}
Mathematica [A] time = 0.643033, size = 269, normalized size = 1.92 \[ \frac{a e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left (\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}}+\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}}\right )}{2 c}+\frac{b \left (\frac{\sqrt{\pi } (2 a-3 b) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b}}+\frac{\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b}}-12 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{a+b \cosh ^{-1}(c x)}+8 c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.
[In]
[Out]
________________________________________________________________________________________
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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]