Optimal. Leaf size=152 \[ -\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^2}-\frac{5 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{15 \sqrt{\sinh ^{-1}(a x)}}{64 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}+\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.336808, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5663, 5758, 5675, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^2}-\frac{5 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{15 \sqrt{\sinh ^{-1}(a x)}}{64 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}+\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5663
Rule 5758
Rule 5675
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \sinh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{1}{4} (5 a) \int \frac{x^2 \sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}+\frac{15}{16} \int x \sqrt{\sinh ^{-1}(a x)} \, dx+\frac{5 \int \frac{\sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{8 a}\\ &=\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{1}{64} (15 a) \int \frac{x^2}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^2}\\ &=\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^2}\\ &=\frac{15 \sqrt{\sinh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a^2}\\ &=\frac{15 \sqrt{\sinh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^2}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a^2}\\ &=\frac{15 \sqrt{\sinh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^2}-\frac{15 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^2}\\ &=\frac{15 \sqrt{\sinh ^{-1}(a x)}}{64 a^2}+\frac{15}{32} x^2 \sqrt{\sinh ^{-1}(a x)}-\frac{5 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{8 a}+\frac{\sinh ^{-1}(a x)^{5/2}}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^2}-\frac{15 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a^2}\\ \end{align*}
Mathematica [A] time = 0.0321814, size = 52, normalized size = 0.34 \[ \frac{\frac{\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-2 \sinh ^{-1}(a x)\right )}{\sqrt{-\sinh ^{-1}(a x)}}+\text{Gamma}\left (\frac{7}{2},2 \sinh ^{-1}(a x)\right )}{32 \sqrt{2} a^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.095, size = 136, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{512\,\sqrt{\pi }{a}^{2}} \left ( -128\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}+160\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1}xa-120\,\sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }{x}^{2}{a}^{2}-64\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }-60\,\sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }+15\,\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +15\,\pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) \right ) } \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 x \operatorname{arsinh}\left (a x\right )^{\frac{5}{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(-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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]