Optimal. Leaf size=111 \[ \frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a}-\frac{\sin ^{-1}(a x)^4}{4 a^2}+\frac{3 \sin ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^4-\frac{3}{2} x^2 \sin ^{-1}(a x)^2+\frac{3 x^2}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.237692, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4627, 4707, 4641, 30} \[ \frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a}-\frac{\sin ^{-1}(a x)^4}{4 a^2}+\frac{3 \sin ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^4-\frac{3}{2} x^2 \sin ^{-1}(a x)^2+\frac{3 x^2}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4627
Rule 4707
Rule 4641
Rule 30
Rubi steps
\begin{align*} \int x \sin ^{-1}(a x)^4 \, dx &=\frac{1}{2} x^2 \sin ^{-1}(a x)^4-(2 a) \int \frac{x^2 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}+\frac{1}{2} x^2 \sin ^{-1}(a x)^4-3 \int x \sin ^{-1}(a x)^2 \, dx-\frac{\int \frac{\sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{a}\\ &=-\frac{3}{2} x^2 \sin ^{-1}(a x)^2+\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{\sin ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^4+(3 a) \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a}-\frac{3}{2} x^2 \sin ^{-1}(a x)^2+\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{\sin ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^4+\frac{3 \int x \, dx}{2}+\frac{3 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{3 x^2}{4}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a}+\frac{3 \sin ^{-1}(a x)^2}{4 a^2}-\frac{3}{2} x^2 \sin ^{-1}(a x)^2+\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a}-\frac{\sin ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0289746, size = 96, normalized size = 0.86 \[ \frac{3 a^2 x^2+\left (2 a^2 x^2-1\right ) \sin ^{-1}(a x)^4+4 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+\left (3-6 a^2 x^2\right ) \sin ^{-1}(a x)^2-6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 117, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ({a}^{2}{x}^{2}-1 \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{2}}+ \left ( \arcsin \left ( ax \right ) \right ) ^{3} \left ( ax\sqrt{-{a}^{2}{x}^{2}+1}+\arcsin \left ( ax \right ) \right ) -{\frac{ \left ( 3\,{a}^{2}{x}^{2}-3 \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{2}}-{\frac{3\,\arcsin \left ( ax \right ) }{2} \left ( ax\sqrt{-{a}^{2}{x}^{2}+1}+\arcsin \left ( ax \right ) \right ) }+{\frac{3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{4}}+{\frac{3\,{a}^{2}{x}^{2}}{4}}-{\frac{3\, \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{4} + 2 \, a \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21133, size = 205, normalized size = 1.85 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \,{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2} + 2 \,{\left (2 \, a x \arcsin \left (a x\right )^{3} - 3 \, a x \arcsin \left (a x\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.34311, size = 104, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asin}^{4}{\left (a x \right )}}{2} - \frac{3 x^{2} \operatorname{asin}^{2}{\left (a x \right )}}{2} + \frac{3 x^{2}}{4} + \frac{x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{a} - \frac{3 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{2 a} - \frac{\operatorname{asin}^{4}{\left (a x \right )}}{4 a^{2}} + \frac{3 \operatorname{asin}^{2}{\left (a x \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.40401, size = 171, normalized size = 1.54 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{a} + \frac{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{2}} + \frac{\arcsin \left (a x\right )^{4}}{4 \, a^{2}} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{2 \, a} - \frac{3 \,{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{2 \, a^{2}} - \frac{3 \, \arcsin \left (a x\right )^{2}}{4 \, a^{2}} + \frac{3 \,{\left (a^{2} x^{2} - 1\right )}}{4 \, a^{2}} + \frac{3}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]