Optimal. Leaf size=26 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3190, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3190
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.00 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 73, normalized size = 2.81 \[ \left [-\frac {\sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \cos \relax (x) - a}{b \cos \relax (x)^{2} + a}\right )}{2 \, a b}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} \cos \relax (x)}{a}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 17, normalized size = 0.65 \[ -\frac {\arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.69 \[ -\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 17, normalized size = 0.65 \[ -\frac {\arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.23, size = 18, normalized size = 0.69 \[ -\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\cos \relax (x)}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 87, normalized size = 3.35 \[ \begin {cases} \frac {\tilde {\infty }}{\cos {\relax (x )}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {1}{b \cos {\relax (x )}} & \text {for}\: a = 0 \\- \frac {\cos {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {i \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \cos {\relax (x )} \right )}}{2 \sqrt {a} b \sqrt {\frac {1}{b}}} - \frac {i \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \cos {\relax (x )} \right )}}{2 \sqrt {a} b \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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