Integrand size = 8, antiderivative size = 75 \[ \int \arccos (a x)^{3/2} \, dx=-\frac {3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{2 a}+x \arccos (a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a} \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4716, 4768, 4720, 3386, 3432} \[ \int \arccos (a x)^{3/2} \, dx=-\frac {3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{2 a}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a}+x \arccos (a x)^{3/2} \]
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Rule 3386
Rule 3432
Rule 4716
Rule 4720
Rule 4768
Rubi steps \begin{align*} \text {integral}& = x \arccos (a x)^{3/2}+\frac {1}{2} (3 a) \int \frac {x \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{2 a}+x \arccos (a x)^{3/2}-\frac {3}{4} \int \frac {1}{\sqrt {\arccos (a x)}} \, dx \\ & = -\frac {3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{2 a}+x \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{4 a} \\ & = -\frac {3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{2 a}+x \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{2 a} \\ & = -\frac {3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{2 a}+x \arccos (a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int \arccos (a x)^{3/2} \, dx=-\frac {\sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},i \arccos (a x)\right )}{2 a \sqrt {\arccos (a x)}} \]
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Time = 0.78 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}-3 \pi \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{4 a \sqrt {\pi }}\) | \(72\) |
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Exception generated. \[ \int \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \arccos (a x)^{3/2} \, dx=\int \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
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Exception generated. \[ \int \arccos (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.59 \[ \int \arccos (a x)^{3/2} \, dx=\frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{16 \, a} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{16 \, a} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{4 \, a} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{4 \, a} \]
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Timed out. \[ \int \arccos (a x)^{3/2} \, dx=\int {\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]
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