Integrand size = 12, antiderivative size = 106 \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5} \]
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Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4732, 4491, 3386, 3432} \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5} \]
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Rule 3386
Rule 3432
Rule 4491
Rule 4732
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^5} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {\sin (x)}{8 \sqrt {x}}+\frac {3 \sin (3 x)}{16 \sqrt {x}}+\frac {\sin (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{a^5} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^5}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^5} \\ & = -\frac {\text {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{8 a^5}-\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{4 a^5}-\frac {3 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{8 a^5} \\ & = -\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.81 \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {-10 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-10 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )-5 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-5 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )-\sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-\sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )}{160 a^5 \sqrt {\arccos (a x)}} \]
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Time = 0.83 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.68
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\sqrt {5}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+10 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{80 a^{5}}\) | \(72\) |
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Exception generated. \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^{4}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]
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Exception generated. \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.31 \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{320 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{320 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{5}} \]
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Timed out. \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^4}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]
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