Integrand size = 12, antiderivative size = 235 \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}+\frac {25 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^5} \]
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Time = 0.28 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4730, 4808, 4732, 4491, 3386, 3432} \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^5}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}} \]
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Rule 3386
Rule 3432
Rule 4491
Rule 4730
Rule 4732
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}-\frac {100}{3} \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx+\frac {16 \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx}{a^2} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}-\frac {16 \text {Subst}\left (\int \left (\frac {\sin (x)}{4 \sqrt {x}}+\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \left (\frac {\sin (x)}{8 \sqrt {x}}+\frac {3 \sin (3 x)}{16 \sqrt {x}}+\frac {\sin (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{3 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}+\frac {25 \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{12 a^5}-\frac {4 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^5}-\frac {4 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{4 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}+\frac {25 \text {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{6 a^5}-\frac {8 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^5}+\frac {25 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{2 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arccos (a x)}}+\frac {20 x^5}{3 \sqrt {\arccos (a x)}}+\frac {25 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{2 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{6 a^5} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.37 \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=-\frac {2 \left (-\sqrt {1-a^2 x^2}-e^{-i \arccos (a x)} \arccos (a x)-e^{i \arccos (a x)} \arccos (a x)+\sqrt {-i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )-5 \arccos (a x) \left (e^{-5 i \arccos (a x)}+e^{5 i \arccos (a x)}-\sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-\sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )\right )-3 \left (3 \arccos (a x) \left (e^{-3 i \arccos (a x)}+e^{3 i \arccos (a x)}-\sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-\sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )+\sin (3 \arccos (a x))\right )-\sin (5 \arccos (a x))}{24 a^5 \arccos (a x)^{3/2}} \]
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Time = 1.01 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {10 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+18 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+4 \arccos \left (a x \right ) a x +10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )+18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+2 \sqrt {-a^{2} x^{2}+1}+3 \sin \left (3 \arccos \left (a x \right )\right )+\sin \left (5 \arccos \left (a x \right )\right )}{24 a^{5} \arccos \left (a x \right )^{\frac {3}{2}}}\) | \(173\) |
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Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]
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