Optimal. Leaf size=60 \[ \frac {x^{1+m} \sinh ^{-1}(a x)}{1+m}-\frac {a x^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-a^2 x^2\right )}{2+3 m+m^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5776, 371}
\begin {gather*} \frac {x^{m+1} \sinh ^{-1}(a x)}{m+1}-\frac {a x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};-a^2 x^2\right )}{m^2+3 m+2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 5776
Rubi steps
\begin {align*} \int x^m \sinh ^{-1}(a x) \, dx &=\frac {x^{1+m} \sinh ^{-1}(a x)}{1+m}-\frac {a \int \frac {x^{1+m}}{\sqrt {1+a^2 x^2}} \, dx}{1+m}\\ &=\frac {x^{1+m} \sinh ^{-1}(a x)}{1+m}-\frac {a x^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 55, normalized size = 0.92 \begin {gather*} \frac {x^{1+m} \left ((2+m) \sinh ^{-1}(a x)-a x \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-a^2 x^2\right )\right )}{(1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 3.44, size = 0, normalized size = 0.00 \[\int x^{m} \arcsinh \left (a x \right )\, dx\]
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]
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 x^{m} \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 = -0.02 \begin {gather*} \int x^m\,\mathrm {asinh}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________