Optimal. Leaf size=47 \[ \frac {b \tanh ^{-1}\left (\frac {a \cos (x)-b \sin (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {\tanh ^{-1}(\sin (x))}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3110, 3770, 3074, 206} \[ \frac {b \tanh ^{-1}\left (\frac {a \cos (x)-b \sin (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}+\frac {\tanh ^{-1}(\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3074
Rule 3110
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tan (x)}{b \cos (x)+a \sin (x)} \, dx &=\int \left (\frac {\sec (x)}{a}-\frac {b}{a (b \cos (x)+a \sin (x))}\right ) \, dx\\ &=\frac {\int \sec (x) \, dx}{a}-\frac {b \int \frac {1}{b \cos (x)+a \sin (x)} \, dx}{a}\\ &=\frac {\tanh ^{-1}(\sin (x))}{a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,a \cos (x)-b \sin (x)\right )}{a}\\ &=\frac {\tanh ^{-1}(\sin (x))}{a}+\frac {b \tanh ^{-1}\left (\frac {a \cos (x)-b \sin (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 76, normalized size = 1.62 \[ \frac {-\frac {2 b \tanh ^{-1}\left (\frac {b \tan \left (\frac {x}{2}\right )-a}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 140, normalized size = 2.98 \[ \frac {\sqrt {a^{2} + b^{2}} b \log \left (\frac {2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - a^{2} - 2 \, b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cos \relax (x) - b \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}}\right ) + {\left (a^{2} + b^{2}\right )} \log \left (\sin \relax (x) + 1\right ) - {\left (a^{2} + b^{2}\right )} \log \left (-\sin \relax (x) + 1\right )}{2 \, {\left (a^{3} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.78, size = 90, normalized size = 1.91 \[ \frac {b \log \left (\frac {{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 63, normalized size = 1.34 \[ -\frac {\ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{a}-\frac {2 b \arctanh \left (\frac {2 b \tan \left (\frac {x}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 98, normalized size = 2.09 \[ \frac {b \log \left (\frac {a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a} + \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 408, normalized size = 8.68 \[ \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a}-\frac {2\,b\,\mathrm {atanh}\left (\frac {64\,b^3}{\sqrt {a^2+b^2}\,\left (128\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {128\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}+\frac {64\,a\,b^3}{a^2+b^2}\right )}-\frac {64\,b^5}{{\left (a^2+b^2\right )}^{3/2}\,\left (128\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {128\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}+\frac {64\,a\,b^3}{a^2+b^2}\right )}+\frac {128\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}\,\left (\frac {64\,a^2\,b^3}{a^2+b^2}+128\,a\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {128\,a\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}\right )}-\frac {128\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\left (a^2+b^2\right )}^{3/2}\,\left (\frac {64\,a^2\,b^3}{a^2+b^2}+128\,a\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {128\,a\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}\right )}+\frac {128\,a\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}\,\left (128\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {128\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}+\frac {64\,a\,b^3}{a^2+b^2}\right )}-\frac {192\,a\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\left (a^2+b^2\right )}^{3/2}\,\left (128\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {128\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}+\frac {64\,a\,b^3}{a^2+b^2}\right )}\right )}{a\,\sqrt {a^2+b^2}} \]
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 \[ \int \frac {\tan {\relax (x )}}{a \sin {\relax (x )} + b \cos {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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