Optimal. Leaf size=89 \[ \frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.18, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4630, 4708, 4642, 4636, 4406, 12, 3305, 3351} \[ \frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4630
Rule 4636
Rule 4642
Rule 4708
Rubi steps
\begin {align*} \int x \cos ^{-1}(a x)^{3/2} \, dx &=\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}-\frac {3}{16} \int \frac {x}{\sqrt {\cos ^{-1}(a x)}} \, dx+\frac {3 \int \frac {\sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^2}\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{16 a^2}\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{8 a}-\frac {\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 64, normalized size = 0.72 \[ \frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )-2 \sqrt {\cos ^{-1}(a x)} \left (3 \sin \left (2 \cos ^{-1}(a x)\right )-4 \cos ^{-1}(a x) \cos \left (2 \cos ^{-1}(a x)\right )\right )}{32 a^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 129, normalized size = 1.45 \[ \frac {3 \, i \sqrt {\arccos \left (a x\right )} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{32 \, a^{2}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{8 \, a^{2}} - \frac {3 \, i \sqrt {\arccos \left (a x\right )} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{32 \, a^{2}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{8 \, a^{2}} - \frac {3 \, \sqrt {\pi } i \operatorname {erf}\left ({\left (i - 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{2} {\left (i - 1\right )}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-{\left (i + 1\right )} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{2} {\left (i - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 64, normalized size = 0.72 \[ \frac {8 \arccos \left (a x \right )^{2} \cos \left (2 \arccos \left (a x \right )\right )+3 \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arccos \left (a x \right ) \sin \left (2 \arccos \left (a x \right )\right )}{32 a^{2} \sqrt {\arccos \left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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