Optimal. Leaf size=70 \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3354, 3352, 3351} \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rubi steps
\begin {align*} \int \cos \left (a+b x^2\right ) \, dx &=\cos (a) \int \cos \left (b x^2\right ) \, dx-\sin (a) \int \sin \left (b x^2\right ) \, dx\\ &=\frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 57, normalized size = 0.81 \[ \frac {\sqrt {\frac {\pi }{2}} \left (\cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 61, normalized size = 0.87 \[ \frac {\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) - \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a)}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.39, size = 95, normalized size = 1.36 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{4 \, {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{4 \, {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 44, normalized size = 0.63 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{2 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.81, size = 48, normalized size = 0.69 \[ -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \relax (a) + \left (i + 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \relax (a) - \left (i - 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )}}{8 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 51, normalized size = 0.73 \[ \frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}\,x}{\sqrt {\pi }}\right )\,\cos \relax (a)}{2\,\sqrt {b}}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}\,x}{\sqrt {\pi }}\right )\,\sin \relax (a)}{2\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 61, normalized size = 0.87 \[ \frac {\sqrt {2} \sqrt {\pi } \left (- \sin {\relax (a )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\relax (a )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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