Integrand size = 12, antiderivative size = 264 \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^5}+\frac {8 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^5} \]
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Time = 0.25 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4729, 4807, 4727, 3386, 3432} \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^5}+\frac {8 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^5}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}} \]
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Rule 3386
Rule 3432
Rule 4727
Rule 4729
Rule 4807
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}+\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}} \, dx}{5 a}-(2 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}} \, dx \\ & = -\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {20}{3} \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx+\frac {16 \int \frac {x^2}{\arcsin (a x)^{3/2}} \, dx}{5 a^2} \\ & = -\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}+\frac {32 \text {Subst}\left (\int \left (-\frac {\sin (x)}{4 \sqrt {x}}+\frac {3 \sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{5 a^5}-\frac {40 \text {Subst}\left (\int \left (-\frac {\sin (x)}{8 \sqrt {x}}+\frac {9 \sin (3 x)}{16 \sqrt {x}}-\frac {5 \sin (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{3 a^5} \\ & = -\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}-\frac {8 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{6 a^5}+\frac {24 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{5 a^5}-\frac {15 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{2 a^5} \\ & = -\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}-\frac {16 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{5 a^5}+\frac {10 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{3 a^5}+\frac {48 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{5 a^5}-\frac {15 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a^5} \\ & = -\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^5}+\frac {8 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^5} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\frac {9 e^{3 i \arcsin (a x)} \left (1+2 i \arcsin (a x)-12 \arcsin (a x)^2\right )+2 e^{i \arcsin (a x)} \left (-3-2 i \arcsin (a x)+4 \arcsin (a x)^2\right )+e^{5 i \arcsin (a x)} \left (-3-10 i \arcsin (a x)+100 \arcsin (a x)^2\right )-8 \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )+e^{-i \arcsin (a x)} \left (-6+4 i \arcsin (a x)+8 \arcsin (a x)^2+8 e^{i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )+108 \sqrt {3} \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )-9 e^{-3 i \arcsin (a x)} \left (-1+2 i \arcsin (a x)+12 \arcsin (a x)^2+12 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )-100 \sqrt {5} \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-5 i \arcsin (a x)\right )+e^{-5 i \arcsin (a x)} \left (-3+10 i \arcsin (a x)+100 \arcsin (a x)^2+100 \sqrt {5} e^{5 i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},5 i \arcsin (a x)\right )\right )}{240 a^5 \arcsin (a x)^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-8 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+108 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-100 \arcsin \left (a x \right )^{2} \cos \left (5 \arcsin \left (a x \right )\right )-4 a x \arcsin \left (a x \right )+18 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-10 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )+6 \sqrt {-a^{2} x^{2}+1}-9 \cos \left (3 \arcsin \left (a x \right )\right )+3 \cos \left (5 \arcsin \left (a x \right )\right )}{120 a^{5} \arcsin \left (a x \right )^{\frac {5}{2}}}\) | \(225\) |
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Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]
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