Optimal. Leaf size=74 \[ \frac {b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )}+\frac {\log (1-\cos (c+d x))}{2 d (a+b)}-\frac {\log (\cos (c+d x)+1)}{2 d (a-b)} \]
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Rubi [A] time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4397, 2721, 801} \[ \frac {b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )}+\frac {\log (1-\cos (c+d x))}{2 d (a+b)}-\frac {\log (\cos (c+d x)+1)}{2 d (a-b)} \]
Antiderivative was successfully verified.
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Rule 801
Rule 2721
Rule 4397
Rubi steps
\begin {align*} \int \frac {1}{a \sin (c+d x)+b \tan (c+d x)} \, dx &=\int \frac {\cot (c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x}{(b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 (a+b) (a-x)}+\frac {1}{2 (a-b) (a+x)}+\frac {b}{(-a+b) (a+b) (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{d}\\ &=\frac {\log (1-\cos (c+d x))}{2 (a+b) d}-\frac {\log (1+\cos (c+d x))}{2 (a-b) d}+\frac {b \log (b+a \cos (c+d x))}{\left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 63, normalized size = 0.85 \[ \frac {(a-b) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\left ((a+b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+b \log (a \cos (c+d x)+b)}{d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 64, normalized size = 0.86 \[ \frac {2 \, b \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a + b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a - b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 100, normalized size = 1.35 \[ \frac {\frac {2 \, b \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{2} - b^{2}} + \frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a + b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 75, normalized size = 1.01 \[ \frac {b \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right ) \left (a -b \right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{d \left (2 a +2 b \right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{d \left (2 a -2 b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 71, normalized size = 0.96 \[ \frac {\frac {b \log \left (a + b - \frac {{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{2} - b^{2}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a + b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 67, normalized size = 0.91 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d\,\left (a+b\right )}+\frac {b\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d\,\left (a^2-b^2\right )} \]
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 \[ \int \frac {1}{a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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