Optimal. Leaf size=89 \[ -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{8 a}-\frac {\text {ArcCos}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {ArcCos}(a x)^{3/2}+\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{32 a^2} \]
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Rubi [A]
time = 0.12, 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 = {4726, 4796,
4738, 4732, 4491, 12, 3386, 3432} \begin {gather*} \frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{8 a}-\frac {\text {ArcCos}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {ArcCos}(a x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4726
Rule 4732
Rule 4738
Rule 4796
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 \text {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 \text {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 \text {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 \text {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 = 0.05, size = 64, normalized size = 0.72 \begin {gather*} \frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )-2 \sqrt {\text {ArcCos}(a x)} (-4 \text {ArcCos}(a x) \cos (2 \text {ArcCos}(a x))+3 \sin (2 \text {ArcCos}(a x)))}{32 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 64, normalized size = 0.72
method | result | size |
default | \(\frac {8 \arccos \left (a x \right )^{2} \cos \left (2 \arccos \left (a x \right )\right )+3 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arccos \left (a x \right ) \sin \left (2 \arccos \left (a x \right )\right )}{32 a^{2} \sqrt {\arccos \left (a x \right )}}\) | \(64\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.48, size = 107, normalized size = 1.20 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{8 \, 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 {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{2}} - \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{128 \, 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 {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{32 \, a^{2}} \end {gather*}
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 \begin {gather*} \int x\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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