Integrand size = 26, antiderivative size = 16 \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=B x+\frac {C \text {arctanh}(\sin (c+d x))}{d} \]
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Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4132, 8, 12, 3855} \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {C \text {arctanh}(\sin (c+d x))}{d}+B x \]
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Rule 8
Rule 12
Rule 3855
Rule 4132
Rubi steps \begin{align*} \text {integral}& = B \int 1 \, dx+\int C \sec (c+d x) \, dx \\ & = B x+C \int \sec (c+d x) \, dx \\ & = B x+\frac {C \text {arctanh}(\sin (c+d x))}{d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=B x+\frac {C \text {arctanh}(\sin (c+d x))}{d} \]
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Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81
method | result | size |
derivativedivides | \(\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \left (d x +c \right )}{d}\) | \(29\) |
default | \(\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \left (d x +c \right )}{d}\) | \(29\) |
parallelrisch | \(\frac {B x d -C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(39\) |
risch | \(B x +\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(42\) |
norman | \(\frac {B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-B x}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, B d x + C \log \left (\sin \left (d x + c\right ) + 1\right ) - C \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
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\[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (16) = 32\).
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.31 \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (d x + c\right )} B + C {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (16) = 32\).
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.69 \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (d x + c\right )} B + C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \]
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Time = 15.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.56 \[ \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,B\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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