Integrand size = 16, antiderivative size = 242 \[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c^3} \]
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Time = 0.33 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4726, 4810, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {1}{3} x^3 \sqrt {a+b \arccos (c x)} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4726
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}+\frac {1}{6} (b c) \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}} \, dx \\ & = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\text {Subst}\left (\int \frac {\cos ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{6 c^3} \\ & = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\text {Subst}\left (\int \left (\frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {3 \cos \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arccos (c x)\right )}{6 c^3} \\ & = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{24 c^3}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{8 c^3} \\ & = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{8 c^3}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{24 c^3}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{8 c^3}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{24 c^3} \\ & = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{4 c^3}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{12 c^3}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{4 c^3}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{12 c^3} \\ & = \frac {1}{3} x^3 \sqrt {a+b \arccos (c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95 \[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=-\frac {i b e^{-\frac {3 i a}{b}} \left (-9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )+\sqrt {3} \left (-\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arccos (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arccos (c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \arccos (c x)}} \]
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Time = 1.89 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {9 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b -\cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b -9 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b +\sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b +18 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +18 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a +6 \arccos \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b +6 \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a}{72 c^{3} \sqrt {a +b \arccos \left (c x \right )}}\) | \(362\) |
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Exception generated. \[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=\int x^{2} \sqrt {a + b \operatorname {acos}{\left (c x \right )}}\, dx \]
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\[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=\int { \sqrt {b \arccos \left (c x\right ) + a} x^{2} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 1057, normalized size of antiderivative = 4.37 \[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^2 \sqrt {a+b \arccos (c x)} \, dx=\int x^2\,\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \]
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