Integrand size = 8, antiderivative size = 76 \[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {4 x}{3 \sqrt {\arccos (a x)}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a} \]
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Time = 0.07 (sec) , antiderivative size = 76, 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 = {4718, 4808, 4720, 3386, 3432} \[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a}+\frac {4 x}{3 \sqrt {\arccos (a x)}} \]
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Rule 3386
Rule 3432
Rule 4718
Rule 4720
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {1}{3} (2 a) \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {4 x}{3 \sqrt {\arccos (a x)}}-\frac {4}{3} \int \frac {1}{\sqrt {\arccos (a x)}} \, dx \\ & = \frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {4 x}{3 \sqrt {\arccos (a x)}}+\frac {4 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a} \\ & = \frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {4 x}{3 \sqrt {\arccos (a x)}}+\frac {8 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a} \\ & = \frac {2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {4 x}{3 \sqrt {\arccos (a x)}}+\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=-\frac {2 \left (-\sqrt {1-a^2 x^2}-e^{-i \arccos (a x)} \arccos (a x)-e^{i \arccos (a x)} \arccos (a x)+\sqrt {-i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )}{3 a \arccos (a x)^{3/2}} \]
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Time = 0.84 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {\sqrt {2}\, \left (4 \arccos \left (a x \right )^{2} \pi \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{3 a \sqrt {\pi }\, \arccos \left (a x \right )^{2}}\) | \(83\) |
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Exception generated. \[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\int \frac {1}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\arccos (a x)^{5/2}} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]
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