Integrand size = 10, antiderivative size = 153 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 \sqrt {b}} \]
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Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3439, 3435, 3433, 3432} \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}} \]
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Rule 3432
Rule 3433
Rule 3435
Rule 3439
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{4} \cos \left (a+b x^2\right )+\frac {1}{4} \cos \left (3 a+3 b x^2\right )\right ) \, dx \\ & = \frac {1}{4} \int \cos \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int \cos \left (a+b x^2\right ) \, dx \\ & = \frac {1}{4} (3 \cos (a)) \int \cos \left (b x^2\right ) \, dx+\frac {1}{4} \cos (3 a) \int \cos \left (3 b x^2\right ) \, dx-\frac {1}{4} (3 \sin (a)) \int \sin \left (b x^2\right ) \, dx-\frac {1}{4} \sin (3 a) \int \sin \left (3 b x^2\right ) \, dx \\ & = \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 \sqrt {b}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{6}} \left (3 \sqrt {3} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\cos (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )-3 \sqrt {3} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)-\operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)\right )}{4 \sqrt {b}} \]
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Time = 0.43 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 \sqrt {b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )-\sin \left (3 a \right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{24 \sqrt {b}}\) | \(101\) |
risch | \(\frac {{\mathrm e}^{-3 i a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, \sqrt {i b}\, x \right )}{48 \sqrt {i b}}+\frac {3 \,{\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{16 \sqrt {i b}}+\frac {{\mathrm e}^{3 i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 i b}\, x \right )}{16 \sqrt {-3 i b}}+\frac {3 \,{\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i b}\, x \right )}{16 \sqrt {-i b}}\) | \(108\) |
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Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {6} \pi \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) + 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) - \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) - 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right )}{24 \, b} \]
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Time = 0.52 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {2} \sqrt {\pi } \left (- \sin {\left (a \right )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (a \right )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{8} + \frac {\sqrt {6} \sqrt {\pi } \left (- \sin {\left (3 a \right )} S\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (3 a \right )} C\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{24} \]
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Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=-\frac {9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (3 \, a\right ) + \left (i + 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3 i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (3 \, a\right ) - \left (i - 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-3 i \, b} x\right )\right )} b^{\frac {3}{2}} - 9 \, \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, \cos \left (a\right ) - \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{96 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.21 \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\frac {i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {6} \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{48 \, \sqrt {b} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{16 \, {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{16 \, {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {6} \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{48 \, \sqrt {b} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
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Timed out. \[ \int \cos ^3\left (a+b x^2\right ) \, dx=\int {\cos \left (b\,x^2+a\right )}^3 \,d x \]
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