Integrand size = 12, antiderivative size = 171 \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7} \]
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Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385, 3433} \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}+\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}} \]
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Rule 3385
Rule 3433
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {5 \cos (x)}{64 \sqrt {x}}-\frac {27 \cos (3 x)}{64 \sqrt {x}}-\frac {25 \cos (5 x)}{64 \sqrt {x}}-\frac {7 \cos (7 x)}{64 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{a^7} \\ & = \frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{32 a^7}-\frac {7 \text {Subst}\left (\int \frac {\cos (7 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{32 a^7}-\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{32 a^7}-\frac {27 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{32 a^7} \\ & = \frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {5 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {7 \text {Subst}\left (\int \cos \left (7 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {25 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {27 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{16 a^7} \\ & = \frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arccos (a x)}\right )}{16 a^7} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.79 \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\frac {i \left (-10 i \sqrt {1-a^2 x^2}+5 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-5 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )+9 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-9 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )+5 \sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-5 \sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )+\sqrt {7} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-7 i \arccos (a x)\right )-\sqrt {7} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},7 i \arccos (a x)\right )-18 i \sin (3 \arccos (a x))-10 i \sin (5 \arccos (a x))-2 i \sin (7 \arccos (a x))\right )}{64 a^7 \sqrt {\arccos (a x)}} \]
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Time = 1.07 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {-\sqrt {2}\, \sqrt {\pi }\, \sqrt {7}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {7}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-9 \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-5 \sqrt {5}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-5 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sqrt {-a^{2} x^{2}+1}+9 \sin \left (3 \arccos \left (a x \right )\right )+5 \sin \left (5 \arccos \left (a x \right )\right )+\sin \left (7 \arccos \left (a x \right )\right )}{32 a^{7} \sqrt {\arccos \left (a x \right )}}\) | \(182\) |
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Exception generated. \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{6}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\int { \frac {x^{6}}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^6}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \]
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