Integrand size = 17, antiderivative size = 26 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {b \log (\tan (c+d x))}{d} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3957, 2913, 2700, 29, 3855} \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\frac {b \log (\tan (c+d x))}{d}-\frac {a \text {arctanh}(\cos (c+d x))}{d} \]
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Rule 29
Rule 2700
Rule 2913
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \csc (c+d x) \sec (c+d x) \, dx \\ & = a \int \csc (c+d x) \, dx+b \int \csc (c+d x) \sec (c+d x) \, dx \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {b \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {b \log (\tan (c+d x))}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b \log (\cos (c+d x))}{d}+\frac {a \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \log (\sin (c+d x))}{d} \]
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Time = 0.57 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+b \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(33\) |
default | \(\frac {a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+b \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(33\) |
parallelrisch | \(\frac {-b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (a +b \right )}{d}\) | \(50\) |
norman | \(\frac {\left (a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(55\) |
risch | \(-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(89\) |
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {2 \, b \log \left (-\cos \left (d x + c\right )\right ) + {\left (a - b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a + b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, d} \]
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\[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \csc {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (a - b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a + b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 2 \, b \log \left (\cos \left (d x + c\right )\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\left (a + b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{2 \, d} \]
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Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\frac {\frac {a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{2}-b\,\ln \left (\cos \left (c+d\,x\right )\right )-\frac {a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{2}+\frac {b\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{2}+\frac {b\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{2}}{d} \]
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