Optimal. Leaf size=49 \[ \frac {\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )}{2 a}+\frac {\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5881, 3389,
2212} \begin {gather*} \frac {\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{2 a}+\frac {\text {Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2212
Rule 3389
Rule 5881
Rubi steps
\begin {align*} \int \cosh ^{-1}(a x)^n \, dx &=\frac {\text {Subst}\left (\int x^n \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {\text {Subst}\left (\int e^{-x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int e^x x^n \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}\\ &=\frac {\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )}{2 a}+\frac {\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 43, normalized size = 0.88 \begin {gather*} \frac {\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )+\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 2.33, size = 40, normalized size = 0.82
method | result | size |
default | \(\frac {\mathrm {arccosh}\left (a x \right )^{n +2} \hypergeom \left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \frac {\mathrm {arccosh}\left (a x \right )^{2}}{4}\right )}{a \left (n +2\right )}\) | \(40\) |
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]
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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {acosh}^{n}{\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 = -0.02 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________