Integrand size = 10, antiderivative size = 70 \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\frac {x}{2}+\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{4 \sqrt {b}} \]
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Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3439, 3435, 3433, 3432} \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } \sin (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}+\frac {x}{2} \]
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Rule 3432
Rule 3433
Rule 3435
Rule 3439
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (2 a+2 b x^2\right )\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{2} \int \cos \left (2 a+2 b x^2\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{2} \cos (2 a) \int \cos \left (2 b x^2\right ) \, dx-\frac {1}{2} \sin (2 a) \int \sin \left (2 b x^2\right ) \, dx \\ & = \frac {x}{2}+\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{4 \sqrt {b}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\frac {2 \sqrt {b} x+\sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )-\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{4 \sqrt {b}} \]
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Time = 0.40 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {x}{2}+\frac {\sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {C}\left (\frac {2 x \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \operatorname {S}\left (\frac {2 x \sqrt {b}}{\sqrt {\pi }}\right )\right )}{4 \sqrt {b}}\) | \(45\) |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{-2 i a} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {i b}\, x \right )}{16 \sqrt {i b}}+\frac {{\mathrm e}^{2 i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-2 i b}\, x \right )}{8 \sqrt {-2 i b}}\) | \(61\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\frac {\pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x \sqrt {\frac {b}{\pi }}\right ) - \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) + 2 \, b x}{4 \, b} \]
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Time = 0.40 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.80 \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\frac {x}{2} + \frac {\sqrt {\pi } \left (- \sin {\left (2 a \right )} S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (2 a \right )} C\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{4} \]
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Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \cos ^2\left (a+b x^2\right ) \, dx=-\frac {4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (2 \, a\right ) + \left (i + 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (2 \, a\right ) - \left (i - 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x\right )\right )} b^{\frac {3}{2}} - 16 \, b^{2} x}{32 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17 \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\frac {1}{2} \, x + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{8 \, \sqrt {b} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{8 \, \sqrt {b} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
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Timed out. \[ \int \cos ^2\left (a+b x^2\right ) \, dx=\int {\cos \left (b\,x^2+a\right )}^2 \,d x \]
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