Optimal. Leaf size=69 \[ -\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.120369, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4678, 8} \[ -\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4620
Rule 4678
Rule 8
Rubi steps
\begin{align*} \int \cos ^{-1}(a x)^4 \, dx &=x \cos ^{-1}(a x)^4+(4 a) \int \frac{x \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4-12 \int \cos ^{-1}(a x)^2 \, dx\\ &=-12 x \cos ^{-1}(a x)^2-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4-(24 a) \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}-12 x \cos ^{-1}(a x)^2-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}-12 x \cos ^{-1}(a x)^2-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0216089, size = 69, normalized size = 1. \[ -\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 67, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arccos \left ( ax \right ) \right ) ^{4}-4\, \left ( \arccos \left ( ax \right ) \right ) ^{3}\sqrt{-{a}^{2}{x}^{2}+1}-12\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}+24\,ax+24\,\arccos \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.48534, size = 100, normalized size = 1.45 \begin{align*} x \arccos \left (a x\right )^{4} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} - 12 \,{\left (\frac{x \arccos \left (a x\right )^{2}}{a} - \frac{2 \,{\left (x + \frac{\sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.38187, size = 149, normalized size = 2.16 \begin{align*} \frac{a x \arccos \left (a x\right )^{4} - 12 \, a x \arccos \left (a x\right )^{2} + 24 \, a x - 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.10413, size = 70, normalized size = 1.01 \begin{align*} \begin{cases} x \operatorname{acos}^{4}{\left (a x \right )} - 12 x \operatorname{acos}^{2}{\left (a x \right )} + 24 x - \frac{4 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{a} + \frac{24 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x}{16} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14171, size = 88, normalized size = 1.28 \begin{align*} x \arccos \left (a x\right )^{4} - 12 \, x \arccos \left (a x\right )^{2} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} + 24 \, x + \frac{24 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]