Integrand size = 10, antiderivative size = 28 \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{2 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4732, 4491, 12, 3386, 3432} \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{2 a^2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4732
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{2 a^2} \\ & = -\frac {\text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^2} \\ & = -\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{2 a^2} \]
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Time = 0.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {\operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{2 a^{2}}\) | \(21\) |
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Exception generated. \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \]
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Exception generated. \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{2}} + \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{8 \, a^{2}} \]
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Timed out. \[ \int \frac {x}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \]
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