Optimal. Leaf size=86 \[ \frac {1}{3} x^3 \sqrt {\text {ArcCos}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4 a^3}-\frac {\sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{12 a^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4726, 4810,
3393, 3385, 3433} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4 a^3}-\frac {\sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{12 a^3}+\frac {1}{3} x^3 \sqrt {\text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3393
Rule 3433
Rule 4726
Rule 4810
Rubi steps
\begin {align*} \int x^2 \sqrt {\cos ^{-1}(a x)} \, dx &=\frac {1}{3} x^3 \sqrt {\cos ^{-1}(a x)}+\frac {1}{6} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=\frac {1}{3} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{6 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{24 a^3}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{12 a^3}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a^3}\\ &=\frac {1}{3} x^3 \sqrt {\cos ^{-1}(a x)}-\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4 a^3}-\frac {\sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{12 a^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 122, normalized size = 1.42 \begin {gather*} \frac {\sqrt {i \text {ArcCos}(a x)} \left (9 \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {3}{2},-i \text {ArcCos}(a x)\right )-9 i \text {ArcCos}(a x) \text {Gamma}\left (\frac {3}{2},i \text {ArcCos}(a x)\right )+\sqrt {3} \left (\sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {3}{2},-3 i \text {ArcCos}(a x)\right )-i \text {ArcCos}(a x) \text {Gamma}\left (\frac {3}{2},3 i \text {ArcCos}(a x)\right )\right )\right )}{72 a^3 \text {ArcCos}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 96, normalized size = 1.12
method | result | size |
default | \(\frac {-\sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-9 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+18 a x \arccos \left (a x \right )+6 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )}{72 a^{3} \sqrt {\arccos \left (a x \right )}}\) | \(96\) |
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} \sqrt {\operatorname {acos}{\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 = 0.49, size = 165, normalized size = 1.92 \begin {gather*} \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 )}{288 \, 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 )}{288 \, a^{3}} + \frac {\left (i + 1\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 )}{32 \, a^{3}} - \frac {\left (i - 1\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 )}{32 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (3 i \, \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-3 i \, \arccos \left (a x\right )\right )}}{24 \, 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\,\sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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