Integrand size = 8, antiderivative size = 105 \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a} \]
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Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4718, 4808, 4810, 3385, 3433} \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a}+\frac {4 x}{15 \arccos (a x)^{3/2}} \]
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Rule 3385
Rule 3433
Rule 4718
Rule 4808
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {1}{5} (2 a) \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx \\ & = \frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {4}{15} \int \frac {1}{\arccos (a x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}-\frac {1}{15} (8 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx \\ & = \frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{15 a} \\ & = \frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {16 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{15 a} \\ & = \frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=-\frac {-6 \sqrt {1-a^2 x^2}-2 i e^{i \arccos (a x)} \arccos (a x) (-i+2 \arccos (a x))-4 (-i \arccos (a x))^{3/2} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+e^{-i \arccos (a x)} \arccos (a x) \left (-2+4 i \arccos (a x)-4 e^{i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )}{15 a \arccos (a x)^{5/2}} \]
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Time = 0.80 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\sqrt {2}\, \left (8 \arccos \left (a x \right )^{3} \pi \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-4 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{15 a \sqrt {\pi }\, \arccos \left (a x \right )^{3}}\) | \(110\) |
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Exception generated. \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\int \frac {1}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]
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