Optimal. Leaf size=42 \[ \frac {\sqrt {\frac {b \cos ^2(x)}{a}+1} F\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {a+b \cos ^2(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3183, 3182} \[ \frac {\sqrt {\frac {b \cos ^2(x)}{a}+1} F\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {a+b \cos ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3182
Rule 3183
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \cos ^2(x)}} \, dx &=\frac {\sqrt {1+\frac {b \cos ^2(x)}{a}} \int \frac {1}{\sqrt {1+\frac {b \cos ^2(x)}{a}}} \, dx}{\sqrt {a+b \cos ^2(x)}}\\ &=\frac {\sqrt {1+\frac {b \cos ^2(x)}{a}} F\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{\sqrt {a+b \cos ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 46, normalized size = 1.10 \[ \frac {\sqrt {\frac {2 a+b \cos (2 x)+b}{a+b}} F\left (x\left |\frac {b}{a+b}\right .\right )}{\sqrt {2 a+b \cos (2 x)+b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \cos \relax (x)^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cos \relax (x)^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 48, normalized size = 1.14 \[ -\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\relax (x )\right )}{a}}\, \EllipticF \left (\cos \relax (x ), \sqrt {-\frac {b}{a}}\right )}{\sin \relax (x ) \sqrt {a +b \left (\cos ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cos \relax (x)^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {b\,{\cos \relax (x)}^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \cos ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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