Integrand size = 19, antiderivative size = 94 \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\log (\cos (c+d x))}{a d}+\frac {\log (1-\sec (c+d x))}{2 (a+b) d}+\frac {\log (1+\sec (c+d x))}{2 (a-b) d}-\frac {b^2 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right ) d} \]
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Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3970, 908} \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b^2 \log (a+b \sec (c+d x))}{a d \left (a^2-b^2\right )}+\frac {\log (1-\sec (c+d x))}{2 d (a+b)}+\frac {\log (\sec (c+d x)+1)}{2 d (a-b)}+\frac {\log (\cos (c+d x))}{a d} \]
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Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \text {Subst}\left (\int \frac {1}{x (a+x) \left (b^2-x^2\right )} \, dx,x,b \sec (c+d x)\right )}{d} \\ & = -\frac {b^2 \text {Subst}\left (\int \left (\frac {1}{2 b^2 (a+b) (b-x)}+\frac {1}{a b^2 x}+\frac {1}{a (a-b) (a+b) (a+x)}-\frac {1}{2 (a-b) b^2 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d} \\ & = \frac {\log (\cos (c+d x))}{a d}+\frac {\log (1-\sec (c+d x))}{2 (a+b) d}+\frac {\log (1+\sec (c+d x))}{2 (a-b) d}-\frac {b^2 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right ) d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87 \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {2 \log (\cos (c+d x))}{a}+\frac {\log (1-\sec (c+d x))}{a+b}+\frac {\log (1+\sec (c+d x))}{a-b}-\frac {2 b^2 \log (a+b \sec (c+d x))}{a^3-a b^2}}{2 d} \]
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Time = 0.62 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {-\frac {b^{2} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right ) \left (a -b \right ) a}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a +2 b}+\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{2 a -2 b}}{d}\) | \(75\) |
default | \(\frac {-\frac {b^{2} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right ) \left (a -b \right ) a}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a +2 b}+\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{2 a -2 b}}{d}\) | \(75\) |
risch | \(\frac {i x}{a}-\frac {i x}{a -b}-\frac {i c}{d \left (a -b \right )}-\frac {i x}{a +b}-\frac {i c}{d \left (a +b \right )}+\frac {2 i b^{2} x}{a \left (a^{2}-b^{2}\right )}+\frac {2 i b^{2} c}{d a \left (a^{2}-b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \left (a -b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \left (a +b \right )}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d a \left (a^{2}-b^{2}\right )}\) | \(193\) |
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {2 \, b^{2} \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{3} - a b^{2}\right )} d} \]
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\[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\cot {\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.72 \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{3} - a b^{2}} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a - b} - \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a + b}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (90) = 180\).
Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.73 \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {a \log \left ({\left | -a - b + \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \right |}\right )}{a^{2} - b^{2}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (\frac {{\left | -2 \, b - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 2 \, {\left | a \right |} \right |}}{{\left | -2 \, b - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 2 \, {\left | a \right |} \right |}}\right )}{{\left (a^{2} - b^{2}\right )} {\left | a \right |}} - \frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a + b}}{2 \, d} \]
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Time = 14.75 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \frac {\cot (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d\,\left (a+b\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}+\frac {b^2\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d\,\left (a\,b^2-a^3\right )} \]
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