Optimal. Leaf size=330 \[ \frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{3 x^4 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{4 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.712764, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5663, 5758, 5717, 5657, 3307, 2180, 2204, 2205, 5669, 5448} \[ \frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{3 x^4 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{4 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5663
Rule 5758
Rule 5717
Rule 5657
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5669
Rule 5448
Rubi steps
\begin{align*} \int x^4 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}-\frac{1}{10} (3 a) \int \frac{x^5 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3}{100} \int \frac{x^4}{\sqrt{\sinh ^{-1}(a x)}} \, dx+\frac{6 \int \frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{25 a}\\ &=\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^4(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}-\frac{4 \int \frac{x \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{25 a^3}-\frac{\int \frac{x^2}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{25 a^2}\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{\cosh (x)}{8 \sqrt{x}}-\frac{3 \cosh (3 x)}{16 \sqrt{x}}+\frac{\cosh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}+\frac{2 \int \frac{1}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{25 a^4}\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{800 a^5}-\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}-\frac{\operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{x}}+\frac{\cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}-\frac{9 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}-\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1600 a^5}+\frac{3 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1600 a^5}+\frac{3 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{800 a^5}+\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{800 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}-\frac{9 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1600 a^5}-\frac{9 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1600 a^5}+\frac{2 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{25 a^5}+\frac{2 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{25 a^5}\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{67 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{1600 a^5}-\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{67 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{1600 a^5}-\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{100 a^5}+\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{100 a^5}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{100 a^5}-\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{100 a^5}\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac{3 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{200 a^5}-\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{200 a^5}-\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}\\ \end{align*}
Mathematica [A] time = 0.11715, size = 152, normalized size = 0.46 \[ \frac{\frac{9 \sqrt{5} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-5 \sinh ^{-1}(a x)\right )}{\sqrt{\sinh ^{-1}(a x)}}+\frac{125 \sqrt{3} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt{-\sinh ^{-1}(a x)}}+\frac{2250 \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-\sinh ^{-1}(a x)\right )}{\sqrt{\sinh ^{-1}(a x)}}-2250 \text{Gamma}\left (\frac{5}{2},\sinh ^{-1}(a x)\right )+125 \sqrt{3} \text{Gamma}\left (\frac{5}{2},3 \sinh ^{-1}(a x)\right )-9 \sqrt{5} \text{Gamma}\left (\frac{5}{2},5 \sinh ^{-1}(a x)\right )}{36000 a^5} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.206, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\it Arcsinh} \left ( ax \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 x^{4} \operatorname{arsinh}\left (a x\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(-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^{4} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]