Optimal. Leaf size=17 \[ \frac {\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5675} \[ \frac {\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5675
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sinh ^{-1}(a x)^{1+n}}{a (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 83, normalized size = 4.88 \[ \frac {\cosh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right ) \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) \sinh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right )}{a n + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.88, size = 29, normalized size = 1.71 \[ \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{n + 1}}{a {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 18, normalized size = 1.06 \[ \frac {\arcsinh \left (a x \right )^{1+n}}{a \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.39, size = 17, normalized size = 1.00 \[ \frac {\operatorname {arsinh}\left (a x\right )^{n + 1}}{a {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.28, size = 33, normalized size = 1.94 \[ \left \{\begin {array}{cl} \frac {\ln \left (\mathrm {asinh}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {asinh}\left (a\,x\right )}^{n+1}}{a\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.87, size = 34, normalized size = 2.00 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge n = -1 \\0^{n} x & \text {for}\: a = 0 \\\frac {\log {\left (\operatorname {asinh}{\left (a x \right )} \right )}}{a} & \text {for}\: n = -1 \\\frac {\operatorname {asinh}{\left (a x \right )} \operatorname {asinh}^{n}{\left (a x \right )}}{a n + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________