Integrand size = 10, antiderivative size = 89 \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a^2} \]
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Time = 0.11 (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 = {4729, 4807, 4731, 4491, 12, 3386, 3432, 4737} \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4729
Rule 4731
Rule 4737
Rule 4807
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (4 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}} \, dx \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {16}{3} \int \frac {x}{\sqrt {\arcsin (a x)}} \, dx \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arcsin (a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {8 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {16 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{3 a^2} \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26 \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 \arcsin (a x) \left (e^{-2 i \arcsin (a x)}+e^{2 i \arcsin (a x)}-\sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )-\sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )\right )+\sin (2 \arcsin (a x))}{3 a^2 \arcsin (a x)^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\frac {8 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+4 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )+\sin \left (2 \arcsin \left (a x \right )\right )}{3 a^{2} \arcsin \left (a x \right )^{\frac {3}{2}}}\) | \(56\) |
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Exception generated. \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=\int \frac {x}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=\int { \frac {x}{\arcsin \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \]
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