Optimal. Leaf size=95 \[ -\frac {3 \sqrt {\text {ArcCos}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {ArcCos}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{64 a^4}-\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{16 a^4} \]
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Rubi [A]
time = 0.14, antiderivative size = 95, 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 (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{64 a^4}-\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{16 a^4}-\frac {3 \sqrt {\text {ArcCos}(a x)}}{32 a^4}+\frac {1}{4} x^4 \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^3 \sqrt {\cos ^{-1}(a x)} \, dx &=\frac {1}{4} x^4 \sqrt {\cos ^{-1}(a x)}+\frac {1}{8} a \int \frac {x^4}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=\frac {1}{4} x^4 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}\\ &=\frac {1}{4} x^4 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac {3 \sqrt {\cos ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^4}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}\\ &=-\frac {3 \sqrt {\cos ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{32 a^4}-\frac {\text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{8 a^4}\\ &=-\frac {3 \sqrt {\cos ^{-1}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\cos ^{-1}(a x)}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{64 a^4}-\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{16 a^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 125, normalized size = 1.32 \begin {gather*} -\frac {\sqrt {\text {ArcCos}(a x)} \left (4 i \sqrt {2} \text {ArcCos}(a x) \text {Gamma}\left (\frac {3}{2},-2 i \text {ArcCos}(a x)\right )+4 \sqrt {2} \sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {3}{2},2 i \text {ArcCos}(a x)\right )+i \text {ArcCos}(a x) \text {Gamma}\left (\frac {3}{2},-4 i \text {ArcCos}(a x)\right )+\sqrt {\text {ArcCos}(a x)^2} \text {Gamma}\left (\frac {3}{2},4 i \text {ArcCos}(a x)\right )\right )}{128 a^4 (-i \text {ArcCos}(a x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 91, normalized size = 0.96
method | result | size |
default | \(\frac {-\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+16 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )+4 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-8 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )}{128 a^{4} \sqrt {\arccos \left (a x \right )}}\) | \(91\) |
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^{3} \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 = 0.48, size = 153, normalized size = 1.61 \begin {gather*} \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{512 \, a^{4}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{512 \, a^{4}} + \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{4}} - \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} \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^3\,\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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