Optimal. Leaf size=201 \[ -\frac {16 x \cos ^{-1}(a x)}{25 a^4}-\frac {8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}-\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3-\frac {6}{125} x^5 \cos ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4628, 4708, 4678, 4620, 261, 266, 43} \[ \frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}-\frac {8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac {16 x \cos ^{-1}(a x)}{25 a^4}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3-\frac {6}{125} x^5 \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 261
Rule 266
Rule 4620
Rule 4628
Rule 4678
Rule 4708
Rubi steps
\begin {align*} \int x^4 \cos ^{-1}(a x)^3 \, dx &=\frac {1}{5} x^5 \cos ^{-1}(a x)^3+\frac {1}{5} (3 a) \int \frac {x^5 \cos ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3-\frac {6}{25} \int x^4 \cos ^{-1}(a x) \, dx+\frac {12 \int \frac {x^3 \cos ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac {6}{125} x^5 \cos ^{-1}(a x)-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3+\frac {8 \int \frac {x \cos ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}-\frac {8 \int x^2 \cos ^{-1}(a x) \, dx}{25 a^2}-\frac {1}{125} (6 a) \int \frac {x^5}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac {6}{125} x^5 \cos ^{-1}(a x)-\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3-\frac {16 \int \cos ^{-1}(a x) \, dx}{25 a^4}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx}{75 a}-\frac {1}{125} (3 a) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {16 x \cos ^{-1}(a x)}{25 a^4}-\frac {8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac {6}{125} x^5 \cos ^{-1}(a x)-\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3-\frac {16 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}-\frac {4 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )}{75 a}-\frac {1}{125} (3 a) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {1-a^2 x}}-\frac {2 \sqrt {1-a^2 x}}{a^4}+\frac {\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=\frac {86 \sqrt {1-a^2 x^2}}{125 a^5}-\frac {4 \left (1-a^2 x^2\right )^{3/2}}{125 a^5}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \cos ^{-1}(a x)}{25 a^4}-\frac {8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac {6}{125} x^5 \cos ^{-1}(a x)-\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3-\frac {4 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{75 a}\\ &=\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}-\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \cos ^{-1}(a x)}{25 a^4}-\frac {8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac {6}{125} x^5 \cos ^{-1}(a x)-\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^3\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 122, normalized size = 0.61 \[ \frac {1125 a^5 x^5 \cos ^{-1}(a x)^3+2 \sqrt {1-a^2 x^2} \left (27 a^4 x^4+136 a^2 x^2+2072\right )-30 a x \left (9 a^4 x^4+20 a^2 x^2+120\right ) \cos ^{-1}(a x)-225 \sqrt {1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cos ^{-1}(a x)^2}{5625 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 104, normalized size = 0.52 \[ \frac {1125 \, a^{5} x^{5} \arccos \left (a x\right )^{3} - 30 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arccos \left (a x\right ) + {\left (54 \, a^{4} x^{4} + 272 \, a^{2} x^{2} - 225 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arccos \left (a x\right )^{2} + 4144\right )} \sqrt {-a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 175, normalized size = 0.87 \[ \frac {1}{5} \, x^{5} \arccos \left (a x\right )^{3} - \frac {6}{125} \, x^{5} \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )^{2}}{25 \, a} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{625 \, a} - \frac {8 \, x^{3} \arccos \left (a x\right )}{75 \, a^{2}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{2}}{25 \, a^{3}} + \frac {272 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{5625 \, a^{3}} - \frac {16 \, x \arccos \left (a x\right )}{25 \, a^{4}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{25 \, a^{5}} + \frac {4144 \, \sqrt {-a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 159, normalized size = 0.79 \[ \frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{3}}{5}-\frac {\arccos \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}+\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arccos \left (a x \right )}{25}-\frac {6 a^{5} x^{5} \arccos \left (a x \right )}{125}+\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arccos \left (a x \right )}{75}+\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.75, size = 171, normalized size = 0.85 \[ \frac {1}{5} \, x^{5} \arccos \left (a x\right )^{3} - \frac {1}{25} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right )^{2} + \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{4}} - \frac {15 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arccos \left (a x\right )}{a^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\mathrm {acos}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.37, size = 202, normalized size = 1.00 \[ \begin {cases} \frac {x^{5} \operatorname {acos}^{3}{\left (a x \right )}}{5} - \frac {6 x^{5} \operatorname {acos}{\left (a x \right )}}{125} - \frac {3 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{25 a} + \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{625 a} - \frac {8 x^{3} \operatorname {acos}{\left (a x \right )}}{75 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{3}} + \frac {272 x^{2} \sqrt {- a^{2} x^{2} + 1}}{5625 a^{3}} - \frac {16 x \operatorname {acos}{\left (a x \right )}}{25 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{5}} + \frac {4144 \sqrt {- a^{2} x^{2} + 1}}{5625 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{5}}{40} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________