Optimal. Leaf size=15 \[ \frac {\tan ^{-1}\left (\frac {b \sin (x)}{a}\right )}{a b} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3190, 205} \[ \frac {\tan ^{-1}\left (\frac {b \sin (x)}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos (x)}{a^2+b^2 \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a^2+b^2 x^2} \, dx,x,\sin (x)\right )\\ &=\frac {\tan ^{-1}\left (\frac {b \sin (x)}{a}\right )}{a b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {b \sin (x)}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (x)}{a^2+b^2 \sin ^2(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.06, size = 15, normalized size = 1.00 \[ \frac {\arctan \left (\frac {b \sin \relax (x)}{a}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 15, normalized size = 1.00 \[ \frac {\arctan \left (\frac {b \sin \relax (x)}{a}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 16, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {b \sin \relax (x )}{a}\right )}{a b}\) | \(16\) |
default | \(\frac {\arctan \left (\frac {b \sin \relax (x )}{a}\right )}{a b}\) | \(16\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{2 i x}-\frac {2 a \,{\mathrm e}^{i x}}{b}-1\right )}{2 b a}-\frac {i \ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}-1\right )}{2 b a}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 15, normalized size = 1.00 \[ \frac {\arctan \left (\frac {b \sin \relax (x)}{a}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 15, normalized size = 1.00 \[ \frac {\mathrm {atan}\left (\frac {b\,\sin \relax (x)}{a}\right )}{a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 31, normalized size = 2.07 \[ \begin {cases} \frac {\tilde {\infty }}{\sin {\relax (x )}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{b^{2} \sin {\relax (x )}} & \text {for}\: a = 0 \\\frac {\sin {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \\\frac {\operatorname {atan}{\left (\frac {b \sin {\relax (x )}}{a} \right )}}{a b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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