Optimal. Leaf size=105 \[ \frac {2 \sqrt {1-a^2 x^2}}{5 a \text {ArcCos}(a x)^{5/2}}+\frac {4 x}{15 \text {ArcCos}(a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\text {ArcCos}(a x)}}+\frac {8 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{15 a} \]
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Rubi [A]
time = 0.11, 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 = {4718, 4808,
4810, 3385, 3433} \begin {gather*} -\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\text {ArcCos}(a x)}}+\frac {2 \sqrt {1-a^2 x^2}}{5 a \text {ArcCos}(a x)^{5/2}}+\frac {8 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{15 a}+\frac {4 x}{15 \text {ArcCos}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4718
Rule 4808
Rule 4810
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 \text {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 \text {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] Result contains complex when optimal does not.
time = 0.38, size = 151, normalized size = 1.44 \begin {gather*} -\frac {-6 \sqrt {1-a^2 x^2}-2 i e^{i \text {ArcCos}(a x)} \text {ArcCos}(a x) (-i+2 \text {ArcCos}(a x))-4 (-i \text {ArcCos}(a x))^{3/2} \text {ArcCos}(a x) \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a x)\right )+e^{-i \text {ArcCos}(a x)} \text {ArcCos}(a x) \left (-2+4 i \text {ArcCos}(a x)-4 e^{i \text {ArcCos}(a x)} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a x)\right )\right )}{15 a \text {ArcCos}(a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 110, normalized size = 1.05
method | result | size |
default | \(\frac {\sqrt {2}\, \left (8 \arccos \left (a x \right )^{3} \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-4 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{15 a \sqrt {\pi }\, \arccos \left (a x \right )^{3}}\) | \(110\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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