Optimal. Leaf size=44 \[ x \sqrt {\text {ArcCos}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a} \]
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Rubi [A]
time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4810,
3385, 3433} \begin {gather*} x \sqrt {\text {ArcCos}(a x)}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4716
Rule 4810
Rubi steps
\begin {align*} \int \sqrt {\cos ^{-1}(a x)} \, dx &=x \sqrt {\cos ^{-1}(a x)}+\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}} \, dx\\ &=x \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 a}\\ &=x \sqrt {\cos ^{-1}(a x)}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a}\\ &=x \sqrt {\cos ^{-1}(a x)}-\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 76, normalized size = 1.73 \begin {gather*} \frac {\sqrt {\text {ArcCos}(a x)} \left (\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {3}{2},-i \text {ArcCos}(a x)\right )+\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {3}{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 = 0.14, size = 49, normalized size = 1.11
method | result | size |
default | \(\frac {-\FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+2 a x \arccos \left (a x \right )}{2 a \sqrt {\arccos \left (a x \right )}}\) | \(49\) |
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 \sqrt {\operatorname {acos}{\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 = 0.45, size = 83, normalized size = 1.89 \begin {gather*} \frac {\left (i + 1\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 )}{8 \, a} - \frac {\left (i - 1\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 )}{8 \, a} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{2 \, 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 = -0.02 \begin {gather*} \int \sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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