Integrand size = 12, antiderivative size = 65 \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{4 a^4} \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4732, 4491, 3386, 3432} \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{4 a^4} \]
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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 ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^4} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{a^4} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{4 a^4} \\ & = -\frac {\text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{4 a^4}-\frac {\text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{2 a^4} \\ & = -\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^4}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{4 a^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.00 \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=-\frac {-2 \sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )-2 \sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )-\sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )-\sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )}{32 a^4 \sqrt {\arccos (a x)}} \]
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Time = 0.75 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \left (\sqrt {2}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+4 \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 a^{4}}\) | \(43\) |
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Exception generated. \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^{3}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25 \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{4}} - \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{16 \, a^{4}} + \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{16 \, a^{4}} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^3}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]
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