Integrand size = 10, antiderivative size = 59 \[ \int x \sqrt {\arccos (a x)} \, dx=-\frac {\sqrt {\arccos (a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^2} \]
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Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4726, 4810, 3393, 3385, 3433} \[ \int x \sqrt {\arccos (a x)} \, dx=-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^2}-\frac {\sqrt {\arccos (a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\arccos (a x)} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 4726
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \sqrt {\arccos (a x)}+\frac {1}{4} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx \\ & = \frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{4 a^2} \\ & = \frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{4 a^2} \\ & = -\frac {\sqrt {\arccos (a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^2} \\ & = -\frac {\sqrt {\arccos (a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{4 a^2} \\ & = -\frac {\sqrt {\arccos (a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int x \sqrt {\arccos (a x)} \, dx=\frac {\frac {1}{4} \sqrt {\arccos (a x)} \cos (2 \arccos (a x))-\frac {1}{8} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{a^2} \]
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Time = 0.67 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {2 \cos \left (2 \arccos \left (a x \right )\right ) \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }-\pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )}{8 a^{2} \sqrt {\pi }}\) | \(43\) |
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Exception generated. \[ \int x \sqrt {\arccos (a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \sqrt {\arccos (a x)} \, dx=\int x \sqrt {\operatorname {acos}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int x \sqrt {\arccos (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.20 \[ \int x \sqrt {\arccos (a x)} \, dx=\frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{2}} - \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{2}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{8 \, a^{2}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{8 \, a^{2}} \]
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Timed out. \[ \int x \sqrt {\arccos (a x)} \, dx=\int x\,\sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \]
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