Optimal. Leaf size=127 \[ -\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}+\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4632, 3304, 3352} \[ -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \text {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}+\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4632
Rubi steps
\begin {align*} \int \frac {x^5}{\cos ^{-1}(a x)^{3/2}} \, dx &=\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \left (-\frac {5 \cos (2 x)}{16 \sqrt {x}}-\frac {\cos (4 x)}{2 \sqrt {x}}-\frac {3 \cos (6 x)}{16 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^6}\\ &=\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (6 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^6}-\frac {5 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^6}\\ &=\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {3 \operatorname {Subst}\left (\int \cos \left (6 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a^6}-\frac {5 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a^6}-\frac {2 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^6}\\ &=\frac {2 x^5 \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^6}-\frac {\sqrt {3 \pi } C\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^6}\\ \end {align*}
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Mathematica [C] time = 0.56, size = 226, normalized size = 1.78 \[ \frac {i \left (-10 i \sin \left (2 \cos ^{-1}(a x)\right )-8 i \sin \left (4 \cos ^{-1}(a x)\right )-2 i \sin \left (6 \cos ^{-1}(a x)\right )+5 \sqrt {2} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \cos ^{-1}(a x)\right )-5 \sqrt {2} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \cos ^{-1}(a x)\right )+8 \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \cos ^{-1}(a x)\right )-8 \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \cos ^{-1}(a x)\right )+\sqrt {6} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 i \cos ^{-1}(a x)\right )-\sqrt {6} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},6 i \cos ^{-1}(a x)\right )\right )}{32 a^6 \sqrt {\cos ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 121, normalized size = 0.95 \[ \frac {-2 \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-8 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-10 \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sin \left (2 \arccos \left (a x \right )\right )+4 \sin \left (4 \arccos \left (a x \right )\right )+\sin \left (6 \arccos \left (a x \right )\right )}{16 a^{6} \sqrt {\arccos \left (a x \right )}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x^5}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{5}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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