Optimal. Leaf size=173 \[ \frac {5^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {\Gamma \left (n+1,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \Gamma \left (n+1,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {5^{-n-1} \Gamma \left (n+1,5 \sinh ^{-1}(a x)\right )}{32 a^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5669, 5448, 3307, 2181} \[ \frac {5^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {\text {Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \text {Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {5^{-n-1} \text {Gamma}\left (n+1,5 \sinh ^{-1}(a x)\right )}{32 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3307
Rule 5448
Rule 5669
Rubi steps
\begin {align*} \int x^4 \sinh ^{-1}(a x)^n \, dx &=\frac {\operatorname {Subst}\left (\int x^n \cosh (x) \sinh ^4(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{8} x^n \cosh (x)-\frac {3}{16} x^n \cosh (3 x)+\frac {1}{16} x^n \cosh (5 x)\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\operatorname {Subst}\left (\int x^n \cosh (5 x) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {\operatorname {Subst}\left (\int x^n \cosh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {3 \operatorname {Subst}\left (\int x^n \cosh (3 x) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac {\operatorname {Subst}\left (\int e^{-5 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\operatorname {Subst}\left (\int e^{5 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\operatorname {Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {\operatorname {Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{-3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {5^{-1-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3^{-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {5^{-1-n} \Gamma \left (1+n,5 \sinh ^{-1}(a x)\right )}{32 a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 145, normalized size = 0.84 \[ \frac {5^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-5 \sinh ^{-1}(a x)\right )-5\ 3^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \sinh ^{-1}(a x)\right )+10 \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\sinh ^{-1}(a x)\right )-10 \Gamma \left (n+1,\sinh ^{-1}(a x)\right )+5\ 3^{-n} \Gamma \left (n+1,3 \sinh ^{-1}(a x)\right )-5^{-n} \Gamma \left (n+1,5 \sinh ^{-1}(a x)\right )}{160 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {arsinh}\left (a x\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arsinh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{4} \arcsinh \left (a x \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arsinh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________