Optimal. Leaf size=119 \[ \frac {32 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^2}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {8 x^2}{15 \cos ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4634, 4720, 4632, 3304, 3352, 4642} \[ \frac {32 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^2}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {8 x^2}{15 \cos ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4632
Rule 4634
Rule 4642
Rule 4720
Rubi steps
\begin {align*} \int \frac {x}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (4 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {8 x^2}{15 \cos ^{-1}(a x)^{3/2}}-\frac {16}{15} \int \frac {x}{\cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {8 x^2}{15 \cos ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {32 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^2}\\ &=\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {8 x^2}{15 \cos ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {64 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{15 a^2}\\ &=\frac {2 x \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {8 x^2}{15 \cos ^{-1}(a x)^{3/2}}-\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {32 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 75, normalized size = 0.63 \[ \frac {32 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )+\frac {4 \cos \left (2 \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^{3/2}}-\frac {\left (16 \cos ^{-1}(a x)^2-3\right ) \sin \left (2 \cos ^{-1}(a x)\right )}{\cos ^{-1}(a x)^{5/2}}}{15 a^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\arccos \left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 73, normalized size = 0.61 \[ -\frac {-32 \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+16 \sin \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-3 \sin \left (2 \arccos \left (a x \right )\right )}{15 a^{2} \arccos \left (a x \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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