Optimal. Leaf size=35 \[ \frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3097, 3133} \[ \frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 3097
Rule 3133
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a \cos (x)+b \sin (x)} \, dx &=\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}\\ &=\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 47, normalized size = 1.34 \[ \frac {2 x (b-i a)-a \log \left ((a \cos (x)+b \sin (x))^2\right )+2 i a \tan ^{-1}(\tan (x))}{2 \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 46, normalized size = 1.31 \[ \frac {2 \, b x - a \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right )}{2 \, {\left (a^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.46, size = 55, normalized size = 1.57 \[ -\frac {a b \log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{2} b + b^{3}} + \frac {b x}{a^{2} + b^{2}} + \frac {a \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 54, normalized size = 1.54 \[ -\frac {a \ln \left (a +b \tan \relax (x )\right )}{a^{2}+b^{2}}+\frac {a \ln \left (1+\tan ^{2}\relax (x )\right )}{2 a^{2}+2 b^{2}}+\frac {b \arctan \left (\tan \relax (x )\right )}{a^{2}+b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 88, normalized size = 2.51 \[ \frac {2 \, b \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2} + b^{2}} - \frac {a \log \left (-a - \frac {2 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} + \frac {a \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 970, normalized size = 27.71 \[ -\frac {a\,\ln \left (a\,\cos \relax (x)+b\,\sin \relax (x)\right )}{a^2+b^2}-\frac {2\,b\,\mathrm {atan}\left (\frac {\left (a^4+2\,a^2\,b^2+b^4\right )\,\left (\mathrm {tan}\left (\frac {x}{2}\right )\,\left (\frac {\left (4\,a^4-13\,a^2\,b^2+b^4\right )\,\left (\frac {b\,\left (64\,a\,b^2+\frac {a\,\left (32\,a^2\,b^2-64\,a^4+\frac {a\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b^3\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^3}+\frac {a\,\left (\frac {b\,\left (32\,a^2\,b^2-64\,a^4+\frac {a\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}\right )}{{\left (4\,a^4+5\,a^2\,b^2+b^4\right )}^2}-\frac {6\,a\,b\,\left (2\,a^2-b^2\right )\,\left (64\,a^2+\frac {a\,\left (64\,a\,b^2+\frac {a\,\left (32\,a^2\,b^2-64\,a^4+\frac {a\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b\,\left (\frac {b\,\left (32\,a^2\,b^2-64\,a^4+\frac {a\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,b\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}-\frac {a\,b^2\,\left (96\,a^3\,b^2+96\,a\,b^4\right )}{{\left (a^2+b^2\right )}^3}\right )}{{\left (4\,a^4+5\,a^2\,b^2+b^4\right )}^2}\right )-\frac {\left (4\,a^4-13\,a^2\,b^2+b^4\right )\,\left (\frac {b\,\left (32\,a^2\,b-\frac {a\,\left (64\,a^3\,b-32\,a\,b^3+\frac {a\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b^3\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^3}-\frac {a\,\left (\frac {b\,\left (64\,a^3\,b-32\,a\,b^3+\frac {a\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}\right )}{{\left (4\,a^4+5\,a^2\,b^2+b^4\right )}^2}+\frac {6\,a\,b\,\left (2\,a^2-b^2\right )\,\left (\frac {a\,\left (32\,a^2\,b-\frac {a\,\left (64\,a^3\,b-32\,a\,b^3+\frac {a\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b\,\left (\frac {b\,\left (64\,a^3\,b-32\,a\,b^3+\frac {a\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {a\,b\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^2}\right )}{a^2+b^2}+\frac {a\,b^2\,\left (96\,a^4\,b+96\,a^2\,b^3\right )}{{\left (a^2+b^2\right )}^3}\right )}{{\left (4\,a^4+5\,a^2\,b^2+b^4\right )}^2}\right )}{32\,a\,b}\right )}{a^2+b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 173, normalized size = 4.94 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\- \frac {i x \sin {\relax (x )}}{- 2 i b \sin {\relax (x )} - 2 b \cos {\relax (x )}} - \frac {x \cos {\relax (x )}}{- 2 i b \sin {\relax (x )} - 2 b \cos {\relax (x )}} + \frac {i \cos {\relax (x )}}{- 2 i b \sin {\relax (x )} - 2 b \cos {\relax (x )}} & \text {for}\: a = - i b \\\frac {i x \sin {\relax (x )}}{2 i b \sin {\relax (x )} - 2 b \cos {\relax (x )}} - \frac {x \cos {\relax (x )}}{2 i b \sin {\relax (x )} - 2 b \cos {\relax (x )}} - \frac {i \cos {\relax (x )}}{2 i b \sin {\relax (x )} - 2 b \cos {\relax (x )}} & \text {for}\: a = i b \\- \frac {\log {\left (\cos {\relax (x )} \right )}}{a} & \text {for}\: b = 0 \\- \frac {a \log {\left (\frac {a \cos {\relax (x )}}{b} + \sin {\relax (x )} \right )}}{a^{2} + b^{2}} + \frac {b x}{a^{2} + b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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