Integrand size = 12, antiderivative size = 163 \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c} \]
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Time = 0.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4717, 4807, 4719, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4717
Rule 4719
Rule 4807
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}-\frac {(2 c) \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}} \, dx}{3 b} \\ & = -\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {4 \int \frac {1}{\sqrt {a+b \arcsin (c x)}} \, dx}{3 b^2} \\ & = -\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {4 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c} \\ & = -\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {\left (4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c}-\frac {\left (4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{3 b^3 c} \\ & = -\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {\left (8 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{3 b^3 c}-\frac {\left (8 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{3 b^3 c} \\ & = -\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\frac {e^{-\frac {i (a+b \arcsin (c x))}{b}} \left (-2 b e^{i \arcsin (c x)} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )-i e^{\frac {i a}{b}} \left (2 a \left (-1+e^{2 i \arcsin (c x)}\right )+b \left (-i-2 \arcsin (c x)+e^{2 i \arcsin (c x)} (-i+2 \arcsin (c x))\right )-2 i b e^{\frac {i (a+b \arcsin (c x))}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )\right )}{3 b^2 c (a+b \arcsin (c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(129)=258\).
Time = 0.06 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.09
method | result | size |
default | \(-\frac {2 \left (2 \arcsin \left (c x \right ) \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b -2 \arcsin \left (c x \right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b +2 \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, a -2 \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, a +2 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +\cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a \right )}{3 c \,b^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\) | \(340\) |
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Exception generated. \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2}} \,d x \]
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