Integrand size = 12, antiderivative size = 159 \[ \int (a+b \arccos (c x))^{3/2} \, dx=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c} \]
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Time = 0.18 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4716, 4768, 4720, 3387, 3386, 3432, 3385, 3433} \[ \int (a+b \arccos (c x))^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4716
Rule 4720
Rule 4768
Rubi steps \begin{align*} \text {integral}& = x (a+b \arccos (c x))^{3/2}+\frac {1}{2} (3 b c) \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}-\frac {1}{4} \left (3 b^2\right ) \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}-\frac {(3 b) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{4 c} \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{4 c}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{4 c} \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{2 c} \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.82 \[ \int (a+b \arccos (c x))^{3/2} \, dx=\frac {\sqrt {b} \left (2 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (-3 \sqrt {1-c^2 x^2}+2 c x \arccos (c x)\right )+\frac {2 i a \sqrt {b} e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )-e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )}{\sqrt {a+b \arccos (c x)}}+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs. \(2(123)=246\).
Time = 2.07 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {-3 \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-3 \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}+4 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b +6 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+4 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}+6 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b}{4 c \sqrt {a +b \arccos \left (c x \right )}}\) | \(278\) |
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Exception generated. \[ \int (a+b \arccos (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (a+b \arccos (c x))^{3/2} \, dx=\int \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (a+b \arccos (c x))^{3/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 993, normalized size of antiderivative = 6.25 \[ \int (a+b \arccos (c x))^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+b \arccos (c x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2} \,d x \]
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