Integrand size = 27, antiderivative size = 27 \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=B x+\frac {C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \sin (c+d x)}{d} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4132, 8, 4130, 3855} \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A \sin (c+d x)}{d}+\frac {C \text {arctanh}(\sin (c+d x))}{d}+B x \]
[In]
[Out]
Rule 8
Rule 3855
Rule 4130
Rule 4132
Rubi steps \begin{align*} \text {integral}& = B \int 1 \, dx+\int \cos (c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = B x+\frac {A \sin (c+d x)}{d}+C \int \sec (c+d x) \, dx \\ & = B x+\frac {C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \sin (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=B x+\frac {C \text {arctanh}(\sin (c+d x))}{d}+\frac {A \cos (d x) \sin (c)}{d}+\frac {A \cos (c) \sin (d x)}{d} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
derivativedivides | \(\frac {A \sin \left (d x +c \right )+B \left (d x +c \right )+C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(37\) |
default | \(\frac {A \sin \left (d x +c \right )+B \left (d x +c \right )+C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(37\) |
parallelrisch | \(\frac {B x d +A \sin \left (d x +c \right )-C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(47\) |
risch | \(B x -\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(74\) |
norman | \(\frac {B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-B x -\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(119\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, B d x + C \log \left (\sin \left (d x + c\right ) + 1\right ) - C \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A \sin \left (d x + c\right )}{2 \, d} \]
[In]
[Out]
\[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (d x + c\right )} B + C {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A \sin \left (d x + c\right )}{2 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (d x + c\right )} B + C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
[In]
[Out]
Time = 14.92 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,B\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,\sin \left (c+d\,x\right )}{d} \]
[In]
[Out]