Optimal. Leaf size=147 \[ -\frac {\sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{6 a}+\frac {1}{3} x^3 \text {ArcCos}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{24 a^3} \]
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Rubi [A]
time = 0.21, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4726, 4796,
4768, 4720, 3386, 3432, 4732, 4491} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{24 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{6 a}-\frac {\sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{3 a^3}+\frac {1}{3} x^3 \text {ArcCos}(a x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3432
Rule 4491
Rule 4720
Rule 4726
Rule 4732
Rule 4768
Rule 4796
Rubi steps
\begin {align*} \int x^2 \cos ^{-1}(a x)^{3/2} \, dx &=\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {1}{2} a \int \frac {x^3 \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}-\frac {1}{12} \int \frac {x^2}{\sqrt {\cos ^{-1}(a x)}} \, dx+\frac {\int \frac {x \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}-\frac {\int \frac {1}{\sqrt {\cos ^{-1}(a x)}} \, dx}{6 a^2}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}+\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^3}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^3}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^3}+\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}+\frac {\text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}\\ &=-\frac {\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{6 a}+\frac {1}{3} x^3 \cos ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{24 a^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 125, normalized size = 0.85 \begin {gather*} -\frac {27 \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-i \text {ArcCos}(a x)\right )+27 \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},i \text {ArcCos}(a x)\right )+\sqrt {3} \left (\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-3 i \text {ArcCos}(a x)\right )+\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},3 i \text {ArcCos}(a x)\right )\right )}{216 a^3 \sqrt {\text {ArcCos}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 130, normalized size = 0.88
method | result | size |
default | \(\frac {36 a x \arccos \left (a x \right )^{2}+\sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+12 \arccos \left (a x \right )^{2} \cos \left (3 \arccos \left (a x \right )\right )+27 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-54 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-6 \arccos \left (a x \right ) \sin \left (3 \arccos \left (a x \right )\right )}{144 a^{3} \sqrt {\arccos \left (a x \right )}}\) | \(130\) |
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^{2} \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 = 0.47, size = 237, normalized size = 1.61 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{576 \, a^{3}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{576 \, a^{3}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{3}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{3}} + \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{48 \, a^{3}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{16 \, a^{3}} - \frac {i \, \sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{48 \, a^{3}} \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^2\,{\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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