Optimal. Leaf size=60 \[ \frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0810922, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4678, 261} \[ \frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4620
Rule 4678
Rule 261
Rubi steps
\begin{align*} \int \cos ^{-1}(a x)^3 \, dx &=x \cos ^{-1}(a x)^3+(3 a) \int \frac{x \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 \int \cos ^{-1}(a x) \, dx\\ &=-6 x \cos ^{-1}(a x)-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-(6 a) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{6 \sqrt{1-a^2 x^2}}{a}-6 x \cos ^{-1}(a x)-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.0186135, size = 60, normalized size = 1. \[ \frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 57, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arccos \left ( ax \right ) \right ) ^{3}-3\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+6\,\sqrt{-{a}^{2}{x}^{2}+1}-6\,ax\arccos \left ( ax \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.4488, size = 80, normalized size = 1.33 \begin{align*} x \arccos \left (a x\right )^{3} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{a} - \frac{6 \,{\left (a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.26834, size = 116, normalized size = 1.93 \begin{align*} \frac{a x \arccos \left (a x\right )^{3} - 6 \, a x \arccos \left (a x\right ) - 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (\arccos \left (a x\right )^{2} - 2\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.572868, size = 60, normalized size = 1. \begin{align*} \begin{cases} x \operatorname{acos}^{3}{\left (a x \right )} - 6 x \operatorname{acos}{\left (a x \right )} - \frac{3 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{a} + \frac{6 \sqrt{- a^{2} x^{2} + 1}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{3} x}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11909, size = 76, normalized size = 1.27 \begin{align*} x \arccos \left (a x\right )^{3} - 6 \, x \arccos \left (a x\right ) - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{a} + \frac{6 \, \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]