Integrand size = 10, antiderivative size = 89 \[ \int x \arccos (a x)^{3/2} \, dx=-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \]
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Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4726, 4796, 4738, 4732, 4491, 12, 3386, 3432} \[ \int x \arccos (a x)^{3/2} \, dx=\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4726
Rule 4732
Rule 4738
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}+\frac {1}{2} x^2 \arccos (a x)^{3/2}-\frac {3}{16} \int \frac {x}{\sqrt {\arccos (a x)}} \, dx+\frac {3 \int \frac {\sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{32 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{8 a}-\frac {\arccos (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^{3/2}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int x \arccos (a x)^{3/2} \, dx=\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )-2 \sqrt {\arccos (a x)} (-4 \arccos (a x) \cos (2 \arccos (a x))+3 \sin (2 \arccos (a x)))}{32 a^2} \]
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Time = 0.79 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {8 \arccos \left (a x \right )^{2} \cos \left (2 \arccos \left (a x \right )\right )+3 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arccos \left (a x \right ) \sin \left (2 \arccos \left (a x \right )\right )}{32 a^{2} \sqrt {\arccos \left (a x \right )}}\) | \(64\) |
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Exception generated. \[ \int x \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \arccos (a x)^{3/2} \, dx=\int x \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
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Exception generated. \[ \int x \arccos (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int x \arccos (a x)^{3/2} \, dx=\frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{8 \, a^{2}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{8 \, a^{2}} + \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{2}} - \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{128 \, a^{2}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{32 \, a^{2}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{32 \, a^{2}} \]
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Timed out. \[ \int x \arccos (a x)^{3/2} \, dx=\int x\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]
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