Optimal. Leaf size=55 \[ \frac {2 \sqrt {1-a^2 x^2}}{a^2}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac {2 x \sin ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4677, 4619, 261} \[ \frac {2 \sqrt {1-a^2 x^2}}{a^2}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac {2 x \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 4619
Rule 4677
Rubi steps
\begin {align*} \int \frac {x \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac {2 \int \sin ^{-1}(a x) \, dx}{a}\\ &=\frac {2 x \sin ^{-1}(a x)}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}-2 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^2}+\frac {2 x \sin ^{-1}(a x)}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 51, normalized size = 0.93 \[ \frac {2 \sqrt {1-a^2 x^2}-\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2+2 a x \sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 35, normalized size = 0.64 \[ \frac {2 \, a x \arcsin \left (a x\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 49, normalized size = 0.89 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2}} + \frac {2 \, {\left (a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 80, normalized size = 1.45 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arcsin \left (a x \right )^{2} x^{2} a^{2}-\arcsin \left (a x \right )^{2}+2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, x a -2 a^{2} x^{2}+2\right )}{a^{2} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.78, size = 49, normalized size = 0.89 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2}} + \frac {2 \, {\left (a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.66, size = 49, normalized size = 0.89 \[ \begin {cases} \frac {2 x \operatorname {asin}{\left (a x \right )}}{a} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{a^{2}} + \frac {2 \sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________