Integrand size = 8, antiderivative size = 44 \[ \int \sqrt {\arccos (a x)} \, dx=x \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \]
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Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4810, 3385, 3433} \[ \int \sqrt {\arccos (a x)} \, dx=x \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \]
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Rule 3385
Rule 3433
Rule 4716
Rule 4810
Rubi steps \begin{align*} \text {integral}& = x \sqrt {\arccos (a x)}+\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx \\ & = x \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{2 a} \\ & = x \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a} \\ & = x \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.57 \[ \int \sqrt {\arccos (a x)} \, dx=\frac {i \left (\sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-i \arccos (a x)\right )-\sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},i \arccos (a x)\right )\right )}{2 a \sqrt {\arccos (a x)}} \]
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Time = 0.88 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {-\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+2 \arccos \left (a x \right ) a x}{2 a \sqrt {\arccos \left (a x \right )}}\) | \(49\) |
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Exception generated. \[ \int \sqrt {\arccos (a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \sqrt {\arccos (a x)} \, dx=\int \sqrt {\operatorname {acos}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \sqrt {\arccos (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.89 \[ \int \sqrt {\arccos (a x)} \, dx=\frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{8 \, a} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{8 \, a} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{2 \, a} \]
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Timed out. \[ \int \sqrt {\arccos (a x)} \, dx=\int \sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \]
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