Optimal. Leaf size=88 \[ -\frac {15}{4} x \sqrt {\text {ArcCos}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{2 a}+x \text {ArcCos}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4 a} \]
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Rubi [A]
time = 0.11, antiderivative size = 88, 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 = {4716, 4768,
4810, 3385, 3433} \begin {gather*} -\frac {5 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4 a}+x \text {ArcCos}(a x)^{5/2}-\frac {15}{4} x \sqrt {\text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4716
Rule 4768
Rule 4810
Rubi steps
\begin {align*} \int \cos ^{-1}(a x)^{5/2} \, dx &=x \cos ^{-1}(a x)^{5/2}+\frac {1}{2} (5 a) \int \frac {x \cos ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}-\frac {15}{4} \int \sqrt {\cos ^{-1}(a x)} \, dx\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}-\frac {1}{8} (15 a) \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a}\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{4 a}\\ &=-\frac {15}{4} x \sqrt {\cos ^{-1}(a x)}-\frac {5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+x \cos ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4 a}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 76, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {\text {ArcCos}(a x)} \left (\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {7}{2},-i \text {ArcCos}(a x)\right )+\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {7}{2},i \text {ArcCos}(a x)\right )\right )}{2 a \sqrt {\text {ArcCos}(a x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 88, normalized size = 1.00
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-4 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +10 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+15 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, a x -15 \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{8 a \sqrt {\pi }}\) | \(88\) |
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 \operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.52, size = 155, normalized size = 1.76 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{4 \, a} - \frac {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (a x\right )\right )}}{4 \, a} - \frac {\left (15 i + 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a} + \frac {\left (15 i - 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{8 \, a} - \frac {15 \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{8 \, a} \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 {\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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