Optimal. Leaf size=121 \[ x \sqrt {a+b \text {ArcCos}(c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c} \]
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Rubi [A]
time = 0.18, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4716, 4810,
3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{c}+x \sqrt {a+b \text {ArcCos}(c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4716
Rule 4810
Rubi steps
\begin {align*} \int \sqrt {a+b \cos ^{-1}(c x)} \, dx &=x \sqrt {a+b \cos ^{-1}(c x)}+\frac {1}{2} (b c) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}} \, dx\\ &=x \sqrt {a+b \cos ^{-1}(c x)}-\frac {b \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{2 c}\\ &=x \sqrt {a+b \cos ^{-1}(c x)}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{2 c}-\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{2 c}\\ &=x \sqrt {a+b \cos ^{-1}(c x)}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{c}\\ &=x \sqrt {a+b \cos ^{-1}(c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 120, normalized size = 0.99 \begin {gather*} -\frac {e^{-\frac {i a}{b}} \sqrt {a+b \text {ArcCos}(c x)} \left (-\frac {\text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcCos}(c x))}{b}\right )}{\sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}}}-\frac {e^{\frac {2 i a}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcCos}(c x))}{b}\right )}{\sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}}}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 186, normalized size = 1.54
method | result | size |
default | \(\frac {-\FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b +\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sin \left (\frac {a}{b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b +2 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a}{2 c \sqrt {a +b \arccos \left (c x \right )}}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 {a + b \operatorname {acos}{\left (c 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 = 0.61, size = 531, normalized size = 4.39 \begin {gather*} -\frac {i \, \sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} + \frac {\sqrt {2} \sqrt {\pi } b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{4 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} + \frac {i \, \sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} + \frac {\sqrt {2} \sqrt {\pi } b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{4 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} + \frac {i \, \sqrt {\pi } a \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{c {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } a \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{c {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} e^{\left (i \, \arccos \left (c x\right )\right )}}{2 \, c} + \frac {\sqrt {b \arccos \left (c x\right ) + a} e^{\left (-i \, \arccos \left (c x\right )\right )}}{2 \, c} \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 \sqrt {a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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