Optimal. Leaf size=69 \[ -\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \text {ArcCos}(a x)+\frac {3 \text {ArcSin}(a x)}{32 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 327, 222}
\begin {gather*} \frac {3 \text {ArcSin}(a x)}{32 a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}+\frac {1}{4} x^4 \text {ArcCos}(a x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 327
Rule 4724
Rubi steps
\begin {align*} \int x^3 \cos ^{-1}(a x) \, dx &=\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {1}{4} a \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3}\\ &=-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {3 \sin ^{-1}(a x)}{32 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 54, normalized size = 0.78 \begin {gather*} \frac {-a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right )+8 a^4 x^4 \text {ArcCos}(a x)+3 \text {ArcSin}(a x)}{32 a^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.00, size = 60, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{16}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (a x \right )}{32}}{a^{4}}\) | \(60\) |
default | \(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{16}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (a x \right )}{32}}{a^{4}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 61, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, x^{4} \arccos \left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.82, size = 48, normalized size = 0.70 \begin {gather*} \frac {{\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{32 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.21, size = 66, normalized size = 0.96 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {acos}{\left (a x \right )}}{4} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{16 a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{32 a^{3}} - \frac {3 \operatorname {acos}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.44, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, x^{4} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{16 \, a} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{32 \, a^{3}} - \frac {3 \, \arccos \left (a x\right )}{32 \, a^{4}} \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 x^3\,\mathrm {acos}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________