Integrand size = 12, antiderivative size = 127 \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^6} \]
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Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385, 3433} \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^6}+\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}} \]
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Rule 3385
Rule 3433
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}+\frac {2 \text {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,\arccos (a x)\right )}{a^6} \\ & = \frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {3 \text {Subst}\left (\int \frac {\cos (6 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^6}-\frac {5 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^6}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^6} \\ & = \frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {3 \text {Subst}\left (\int \cos \left (6 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{4 a^6}-\frac {5 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{4 a^6}-\frac {2 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^6} \\ & = \frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^6} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.78 \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\frac {i \left (5 \sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )-5 \sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )+8 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )-8 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )+\sqrt {6} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-6 i \arccos (a x)\right )-\sqrt {6} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},6 i \arccos (a x)\right )-10 i \sin (2 \arccos (a x))-8 i \sin (4 \arccos (a x))-2 i \sin (6 \arccos (a x))\right )}{32 a^6 \sqrt {\arccos (a x)}} \]
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Time = 0.94 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {-8 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-2 \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-10 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sin \left (2 \arccos \left (a x \right )\right )+4 \sin \left (4 \arccos \left (a x \right )\right )+\sin \left (6 \arccos \left (a x \right )\right )}{16 a^{6} \sqrt {\arccos \left (a x \right )}}\) | \(121\) |
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Exception generated. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{5}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^5}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]
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