Integrand size = 12, antiderivative size = 179 \[ \int (a+b \arccos (c x))^{5/2} \, dx=-\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c} \]
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Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4716, 4768, 4810, 3387, 3386, 3432, 3385, 3433} \[ \int (a+b \arccos (c x))^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{4 c}-\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4716
Rule 4768
Rule 4810
Rubi steps \begin{align*} \text {integral}& = x (a+b \arccos (c x))^{5/2}+\frac {1}{2} (5 b c) \int \frac {x (a+b \arccos (c x))^{3/2}}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}-\frac {1}{4} \left (15 b^2\right ) \int \sqrt {a+b \arccos (c x)} \, dx \\ & = -\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}-\frac {1}{8} \left (15 b^3 c\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}} \, dx \\ & = -\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {\left (15 b^2\right ) \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} \\ & = -\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {\left (15 b^2 \cos \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 )}{8 c}+\frac {\left (15 b^2 \sin \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 )}{8 c} \\ & = -\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {\left (15 b^2 \cos \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 )}{4 c}+\frac {\left (15 b^2 \sin \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 )}{4 c} \\ & = -\frac {15}{4} b^2 x \sqrt {a+b \arccos (c x)}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}{2 c}+x (a+b \arccos (c x))^{5/2}+\frac {15 b^{5/2} \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}+\frac {15 b^{5/2} \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} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.40 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.08 \[ \int (a+b \arccos (c x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arccos (c x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arccos (c x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arccos (c x)) \left (5 \left (3 b c x+2 a \sqrt {1-c^2 x^2}\right )+\left (-8 a c x+10 b \sqrt {1-c^2 x^2}\right ) \arccos (c x)-4 b c x \arccos (c x)^2\right )-2 i a^2 \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+2 i a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )\right )}{16 c \sqrt {a+b \arccos (c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).
Time = 2.08 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.24
method | result | size |
default | \(\frac {15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, b^{3}-15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, b^{3}+8 \arccos \left (c x \right )^{3} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+20 \arccos \left (c x \right )^{2} \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b -30 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+40 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+8 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{3}-30 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+20 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b}{8 c \sqrt {a +b \arccos \left (c x \right )}}\) | \(401\) |
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Exception generated. \[ \int (a+b \arccos (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (a+b \arccos (c x))^{5/2} \, dx=\int \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int (a+b \arccos (c x))^{5/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.58 (sec) , antiderivative size = 1177, normalized size of antiderivative = 6.58 \[ \int (a+b \arccos (c x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+b \arccos (c x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \]
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