Optimal. Leaf size=70 \[ \frac {x}{2}+\frac {\sqrt {\pi } \cos (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{4 \sqrt {b}} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3439, 3435,
3433, 3432} \begin {gather*} \frac {\sqrt {\pi } \cos (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } \sin (2 a) S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}+\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3435
Rule 3439
Rubi steps
\begin {align*} \int \cos ^2\left (a+b x^2\right ) \, dx &=\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) C\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 67, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {b} x+\sqrt {\pi } \cos (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )-\sqrt {\pi } S\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 45, normalized size = 0.64
method | result | size |
default | \(\frac {x}{2}+\frac {\sqrt {\pi }\, \left (\cos \left (2 a \right ) \FresnelC \left (\frac {2 x \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \mathrm {S}\left (\frac {2 x \sqrt {b}}{\sqrt {\pi }}\right )\right )}{4 \sqrt {b}}\) | \(45\) |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{-2 i a} \sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {i b}\, x \right )}{16 \sqrt {i b}}+\frac {{\mathrm e}^{2 i a} \sqrt {\pi }\, \erf \left (\sqrt {-2 i b}\, x \right )}{8 \sqrt {-2 i b}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 70, normalized size = 1.00 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 59, normalized size = 0.84 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 56, normalized size = 0.80 \begin {gather*} \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} \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 = 0.41, size = 82, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, x - \frac {\sqrt {\pi } \operatorname {erf}\left (-\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 {\sqrt {\pi } \operatorname {erf}\left (-\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 )}} \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 {\cos \left (b\,x^2+a\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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