Integrand size = 12, antiderivative size = 121 \[ \int \sqrt {a+b \arccos (c x)} \, dx=x \sqrt {a+b \arccos (c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c} \]
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Time = 0.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4716, 4810, 3387, 3386, 3432, 3385, 3433} \[ \int \sqrt {a+b \arccos (c x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{c}+x \sqrt {a+b \arccos (c x)} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4716
Rule 4810
Rubi steps \begin{align*} \text {integral}& = x \sqrt {a+b \arccos (c x)}+\frac {1}{2} (b c) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}} \, dx \\ & = x \sqrt {a+b \arccos (c x)}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{2 c} \\ & = x \sqrt {a+b \arccos (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{2 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{2 c} \\ & = x \sqrt {a+b \arccos (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 \arccos (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 \arccos (c x)}\right )}{c} \\ & = x \sqrt {a+b \arccos (c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \sqrt {a+b \arccos (c x)} \, dx=-\frac {i b e^{-\frac {i a}{b}} \left (-\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )}{2 c \sqrt {a+b \arccos (c x)}} \]
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Time = 1.84 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.54
method | result | size |
default | \(\frac {-\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b +\sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, b +2 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a}{2 c \sqrt {a +b \arccos \left (c x \right )}}\) | \(186\) |
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Exception generated. \[ \int \sqrt {a+b \arccos (c x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \sqrt {a+b \arccos (c x)} \, dx=\int \sqrt {a + b \operatorname {acos}{\left (c x \right )}}\, dx \]
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\[ \int \sqrt {a+b \arccos (c x)} \, dx=\int { \sqrt {b \arccos \left (c x\right ) + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 531, normalized size of antiderivative = 4.39 \[ \int \sqrt {a+b \arccos (c x)} \, dx=-\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} \]
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Timed out. \[ \int \sqrt {a+b \arccos (c x)} \, dx=\int \sqrt {a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \]
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