Optimal. Leaf size=91 \[ \frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^4}-\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{a^4} \]
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Rubi [A]
time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385,
3433} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^4}-\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{a^4}+\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4728
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)^{3/2}} \, dx &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {\cos (2 x)}{2 \sqrt {x}}-\frac {\cos (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {2 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^4}-\frac {2 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^4}-\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 154, normalized size = 1.69 \begin {gather*} \frac {i \sqrt {2} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcCos}(a x)\right )-i \sqrt {2} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcCos}(a x)\right )+i \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcCos}(a x)\right )-i \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcCos}(a x)\right )+2 \sin (2 \text {ArcCos}(a x))+\sin (4 \text {ArcCos}(a x))}{4 a^4 \sqrt {\text {ArcCos}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 81, normalized size = 0.89
method | result | size |
default | \(\frac {-2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-4 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+2 \sin \left (2 \arccos \left (a x \right )\right )+\sin \left (4 \arccos \left (a x \right )\right )}{4 a^{4} \sqrt {\arccos \left (a x \right )}}\) | \(81\) |
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 \frac {x^{3}}{\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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\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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