Optimal. Leaf size=89 \[ \frac {x \sqrt {1-a^2 x^2}}{4 a^2}-\frac {\text {ArcSin}(a x)}{4 a^3}+\frac {x^2 \text {ArcSin}(a x)}{2 a}-\frac {x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{2 a^2}+\frac {\text {ArcSin}(a x)^3}{6 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4795, 4737,
4723, 327, 222} \begin {gather*} \frac {\text {ArcSin}(a x)^3}{6 a^3}-\frac {\text {ArcSin}(a x)}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{2 a^2}+\frac {x \sqrt {1-a^2 x^2}}{4 a^2}+\frac {x^2 \text {ArcSin}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4795
Rubi steps
\begin {align*} \int \frac {x^2 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac {\int \frac {\sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}+\frac {\int x \sin ^{-1}(a x) \, dx}{a}\\ &=\frac {x^2 \sin ^{-1}(a x)}{2 a}-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac {\sin ^{-1}(a x)^3}{6 a^3}-\frac {1}{2} \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x \sqrt {1-a^2 x^2}}{4 a^2}+\frac {x^2 \sin ^{-1}(a x)}{2 a}-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac {\sin ^{-1}(a x)^3}{6 a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{4 a^2}-\frac {\sin ^{-1}(a x)}{4 a^3}+\frac {x^2 \sin ^{-1}(a x)}{2 a}-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac {\sin ^{-1}(a x)^3}{6 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 73, normalized size = 0.82 \begin {gather*} \frac {3 a x \sqrt {1-a^2 x^2}+\left (-3+6 a^2 x^2\right ) \text {ArcSin}(a x)-6 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2+2 \text {ArcSin}(a x)^3}{12 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 71, normalized size = 0.80
method | result | size |
default | \(\frac {-6 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +6 a^{2} x^{2} \arcsin \left (a x \right )+2 \arcsin \left (a x \right )^{3}+3 a x \sqrt {-a^{2} x^{2}+1}-3 \arcsin \left (a x \right )}{12 a^{3}}\) | \(71\) |
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 [A]
time = 2.24, size = 59, normalized size = 0.66 \begin {gather*} \frac {2 \, \arcsin \left (a x\right )^{3} + 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{12 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.36, size = 78, normalized size = 0.88 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {asin}{\left (a x \right )}}{2 a} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{2 a^{2}} + \frac {x \sqrt {- a^{2} x^{2} + 1}}{4 a^{2}} + \frac {\operatorname {asin}^{3}{\left (a x \right )}}{6 a^{3}} - \frac {\operatorname {asin}{\left (a x \right )}}{4 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.45, size = 81, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{3}}{6 \, a^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{4 \, a^{2}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{2 \, a^{3}} + \frac {\arcsin \left (a x\right )}{4 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________