Integrand size = 10, antiderivative size = 119 \[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4}{15 a^2 \arccos (a x)^{3/2}}+\frac {8 x^2}{15 \arccos (a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{15 a^2} \]
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Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4730, 4808, 4728, 3385, 3433, 4738} \[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{15 a^2}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4}{15 a^2 \arccos (a x)^{3/2}}+\frac {8 x^2}{15 \arccos (a x)^{3/2}} \]
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Rule 3385
Rule 3433
Rule 4728
Rule 4730
Rule 4738
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (4 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4}{15 a^2 \arccos (a x)^{3/2}}+\frac {8 x^2}{15 \arccos (a x)^{3/2}}-\frac {16}{15} \int \frac {x}{\arccos (a x)^{3/2}} \, dx \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4}{15 a^2 \arccos (a x)^{3/2}}+\frac {8 x^2}{15 \arccos (a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {32 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{15 a^2} \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4}{15 a^2 \arccos (a x)^{3/2}}+\frac {8 x^2}{15 \arccos (a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {64 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{15 a^2} \\ & = \frac {2 x \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {4}{15 a^2 \arccos (a x)^{3/2}}+\frac {8 x^2}{15 \arccos (a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{15 a^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.63 \[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\frac {\frac {4 \cos (2 \arccos (a x))}{\arccos (a x)^{3/2}}+32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )-\frac {\left (-3+16 \arccos (a x)^2\right ) \sin (2 \arccos (a x))}{\arccos (a x)^{5/2}}}{15 a^2} \]
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Time = 0.83 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61
method | result | size |
default | \(-\frac {-32 \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+16 \sin \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-3 \sin \left (2 \arccos \left (a x \right )\right )}{15 a^{2} \arccos \left (a x \right )^{\frac {5}{2}}}\) | \(73\) |
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Exception generated. \[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\int \frac {x}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\arccos (a x)^{7/2}} \, dx=\int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]
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