Optimal. Leaf size=182 \[ \frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5777, 5819,
3393, 3389, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}+\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rule 5777
Rule 5819
Rubi steps
\begin {align*} \int x^4 \sqrt {\sinh ^{-1}(a x)} \, dx &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh ^5(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}+\frac {i \text {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 \sqrt {x}}-\frac {5 i \sinh (3 x)}{16 \sqrt {x}}+\frac {i \sinh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{160 a^5}+\frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{320 a^5}-\frac {\text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{320 a^5}-\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^5}+\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}+\frac {\text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{160 a^5}-\frac {\text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{160 a^5}-\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{16 a^5}\\ &=\frac {1}{5} x^5 \sqrt {\sinh ^{-1}(a x)}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 161, normalized size = 0.88 \begin {gather*} \frac {\frac {\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-5 \sinh ^{-1}(a x)\right )}{160 \sqrt {5} \sqrt {-\sinh ^{-1}(a x)}}-\frac {\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-3 \sinh ^{-1}(a x)\right )}{32 \sqrt {3} \sqrt {-\sinh ^{-1}(a x)}}+\frac {\sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-\sinh ^{-1}(a x)\right )}{16 \sqrt {-\sinh ^{-1}(a x)}}-\frac {1}{16} \Gamma \left (\frac {3}{2},\sinh ^{-1}(a x)\right )+\frac {\Gamma \left (\frac {3}{2},3 \sinh ^{-1}(a x)\right )}{32 \sqrt {3}}-\frac {\Gamma \left (\frac {3}{2},5 \sinh ^{-1}(a x)\right )}{160 \sqrt {5}}}{a^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 6.61, size = 0, normalized size = 0.00 \[\int x^{4} \sqrt {\arcsinh \left (a x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\sqrt {\mathrm {asinh}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________