Integrand size = 19, antiderivative size = 26 \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=-\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (x)}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4810, 3385, 3433} \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=-\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (x)}\right ) \]
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Rule 3385
Rule 3433
Rule 4810
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (x)\right ) \\ & = -\left (2 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (x)}\right )\right ) \\ & = -\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (x)}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=\frac {i \left (\sqrt {-i \arccos (x)} \Gamma \left (\frac {1}{2},-i \arccos (x)\right )-\sqrt {i \arccos (x)} \Gamma \left (\frac {1}{2},i \arccos (x)\right )\right )}{2 \sqrt {\arccos (x)}} \]
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Time = 1.62 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\) | \(21\) |
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Exception generated. \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=\int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {\operatorname {acos}{\left (x \right )}}}\, dx \]
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Exception generated. \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (x\right )}\right ) - \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (x\right )}\right ) \]
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Timed out. \[ \int \frac {x}{\sqrt {1-x^2} \sqrt {\arccos (x)}} \, dx=\int \frac {x}{\sqrt {\mathrm {acos}\left (x\right )}\,\sqrt {1-x^2}} \,d x \]
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