Optimal. Leaf size=25 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 25, 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} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 78, normalized size = 3.12 \[ \left [-\frac {\sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right )}{2 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \relax (x)}{a}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 16, normalized size = 0.64 \[ \frac {\arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 17, normalized size = 0.68 \[ \frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 16, normalized size = 0.64 \[ \frac {\arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.64, size = 17, normalized size = 0.68 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \relax (x)}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 87, normalized size = 3.48 \[ \begin {cases} \frac {\tilde {\infty }}{\sin {\relax (x )}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{b \sin {\relax (x )}} & \text {for}\: a = 0 \\\frac {\sin {\relax (x )}}{a} & \text {for}\: b = 0 \\- \frac {i \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sin {\relax (x )} \right )}}{2 \sqrt {a} b \sqrt {\frac {1}{b}}} + \frac {i \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sin {\relax (x )} \right )}}{2 \sqrt {a} b \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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