Optimal. Leaf size=105 \[ -\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {8 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4622, 4720, 4724, 3304, 3352} \[ -\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {8 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4622
Rule 4720
Rule 4724
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {1}{5} (2 a) \int \frac {x}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}}-\frac {4}{15} \int \frac {1}{\cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}-\frac {1}{15} (8 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{15 a}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac {4 x}{15 \cos ^{-1}(a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {8 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a}\\ \end {align*}
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Mathematica [C] time = 1.17, size = 151, normalized size = 1.44 \[ -\frac {-6 \sqrt {1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (2 \cos ^{-1}(a x)-i\right )-4 \cos ^{-1}(a x) \left (-i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a x)\right )+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left (4 i \cos ^{-1}(a x)-4 e^{i \cos ^{-1}(a x)} \left (i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a x)\right )-2\right )}{15 a \cos ^{-1}(a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int \frac {1}{\arccos \left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 110, normalized size = 1.05 \[ \frac {\sqrt {2}\, \left (8 \pi \arccos \left (a x \right )^{3} \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-4 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x a +3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{15 a \sqrt {\pi }\, \arccos \left (a x \right )^{3}} \]
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 \[ \int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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