Optimal. Leaf size=55 \[ \frac {(a+b x) \sqrt {\text {ArcCos}(a+b x)}}{b}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{b} \]
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Rubi [A]
time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4888, 4716,
4810, 3385, 3433} \begin {gather*} \frac {(a+b x) \sqrt {\text {ArcCos}(a+b x)}}{b}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4716
Rule 4810
Rule 4888
Rubi steps
\begin {align*} \int \sqrt {\cos ^{-1}(a+b x)} \, dx &=\frac {\text {Subst}\left (\int \sqrt {\cos ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sqrt {\cos ^{-1}(a+b x)}}{b}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{2 b}\\ &=\frac {(a+b x) \sqrt {\cos ^{-1}(a+b x)}}{b}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {(a+b x) \sqrt {\cos ^{-1}(a+b x)}}{b}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a+b x)}\right )}{b}\\ &=\frac {(a+b x) \sqrt {\cos ^{-1}(a+b x)}}{b}-\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 90, normalized size = 1.64 \begin {gather*} -\frac {-\frac {\sqrt {\text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {3}{2},-i \text {ArcCos}(a+b x)\right )}{2 \sqrt {-i \text {ArcCos}(a+b x)}}-\frac {\sqrt {\text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {3}{2},i \text {ArcCos}(a+b x)\right )}{2 \sqrt {i \text {ArcCos}(a+b x)}}}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 66, normalized size = 1.20
method | result | size |
default | \(\frac {-\FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }+2 \arccos \left (b x +a \right ) b x +2 \arccos \left (b x +a \right ) a}{2 b \sqrt {\arccos \left (b x +a \right )}}\) | \(66\) |
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 + b 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 = 0.44, size = 95, normalized size = 1.73 \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 (b x + a\right )}\right )}{8 \, b} - \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 (b x + a\right )}\right )}{8 \, b} + \frac {\sqrt {\arccos \left (b x + a\right )} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {\sqrt {\arccos \left (b x + a\right )} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{2 \, b} \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+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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