Integrand size = 10, antiderivative size = 89 \[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}+\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{3 a^2} \]
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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 = {4730, 4808, 4732, 4491, 12, 3386, 3432, 4738} \[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{3 a^2}+\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4730
Rule 4732
Rule 4738
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (4 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}-\frac {16}{3} \int \frac {x}{\sqrt {\arccos (a x)}} \, dx \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}+\frac {16 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^2} \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}+\frac {16 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^2} \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}+\frac {8 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^2} \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}+\frac {16 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^2} \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arccos (a x)}}+\frac {8 x^2}{3 \sqrt {\arccos (a x)}}+\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{3 a^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )+\frac {4 \arccos (a x) \cos (2 \arccos (a x))+\sin (2 \arccos (a x))}{\arccos (a x)^{3/2}}}{3 a^2} \]
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Time = 0.78 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {8 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )+\sin \left (2 \arccos \left (a x \right )\right )}{3 a^{2} \arccos \left (a x \right )^{\frac {3}{2}}}\) | \(56\) |
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Exception generated. \[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\int \frac {x}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\arccos (a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]
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