Integrand size = 12, antiderivative size = 125 \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3} \]
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Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4730, 4808, 4732, 4491, 3386, 3432, 4720} \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}} \]
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Rule 3386
Rule 3432
Rule 4491
Rule 4720
Rule 4730
Rule 4732
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx}{3 a}+(2 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}} \, dx \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}-12 \int \frac {x^2}{\sqrt {\arccos (a x)}} \, dx+\frac {8 \int \frac {1}{\sqrt {\arccos (a x)}} \, dx}{3 a^2} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}-\frac {8 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^3}+\frac {12 \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}-\frac {16 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {12 \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^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}-\frac {8 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^3}+\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{a^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}-\frac {8 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {6 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^3}+\frac {6 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.76 \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=-\frac {-\sqrt {1-a^2 x^2}-e^{-i \arccos (a x)} \arccos (a x)-e^{i \arccos (a x)} \arccos (a x)+\sqrt {-i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},i \arccos (a x)\right )-3 \arccos (a x) \left (e^{-3 i \arccos (a x)}+e^{3 i \arccos (a x)}-\sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-\sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )-\sin (3 \arccos (a x))}{6 a^3 \arccos (a x)^{3/2}} \]
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Time = 0.86 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {6 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+2 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+2 \arccos \left (a x \right ) a x +6 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+\sqrt {-a^{2} x^{2}+1}+\sin \left (3 \arccos \left (a x \right )\right )}{6 a^{3} \arccos \left (a x \right )^{\frac {3}{2}}}\) | \(115\) |
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Exception generated. \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^{2}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]
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