Integrand size = 12, antiderivative size = 97 \[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3} \]
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Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385, 3433} \[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}} \]
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Rule 3385
Rule 3433
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {\cos (x)}{4 \sqrt {x}}-\frac {3 \cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{a^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{2 a^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^3}-\frac {3 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^3} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.64 \[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\frac {i \left (-2 i \sqrt {1-a^2 x^2}+\sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-\sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )+\sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-\sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )-2 i \sin (3 \arccos (a x))\right )}{4 a^3 \sqrt {\arccos (a x)}} \]
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Time = 0.83 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {-\sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+\sqrt {-a^{2} x^{2}+1}+\sin \left (3 \arccos \left (a x \right )\right )}{2 a^{3} \sqrt {\arccos \left (a x \right )}}\) | \(94\) |
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Exception generated. \[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{2}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]
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