Optimal. Leaf size=56 \[ x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1+n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5874, 12, 371}
\begin {gather*} x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{n+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 371
Rule 5874
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (a x^n\right ) \, dx &=x \sinh ^{-1}\left (a x^n\right )-\int \frac {a n x^n}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=x \sinh ^{-1}\left (a x^n\right )-(a n) \int \frac {x^n}{\sqrt {1+a^2 x^{2 n}}} \, dx\\ &=x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1+n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 56, normalized size = 1.00 \begin {gather*} x \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1+n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \arcsinh \left (a \,x^{n}\right )\, dx\]
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 \operatorname {asinh}{\left (a x^{n} \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 {asinh}\left (a\,x^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________