Integrand size = 37, antiderivative size = 69 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {1}{2} (2 A b+2 a B+b C) x+\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {b C \cos (c+d x) \sin (c+d x)}{2 d} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {3112, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {1}{2} x (2 a B+2 A b+b C)+\frac {(a C+b B) \sin (c+d x)}{d}+\frac {b C \sin (c+d x) \cos (c+d x)}{2 d} \]
[In]
[Out]
Rule 2814
Rule 3102
Rule 3112
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b C \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a A+(2 A b+2 a B+b C) \cos (c+d x)+2 (b B+a C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {(b B+a C) \sin (c+d x)}{d}+\frac {b C \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \int (2 a A+(2 A b+2 a B+b C) \cos (c+d x)) \sec (c+d x) \, dx \\ & = \frac {1}{2} (2 A b+2 a B+b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {b C \cos (c+d x) \sin (c+d x)}{2 d}+(a A) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} (2 A b+2 a B+b C) x+\frac {a A \text {arctanh}(\sin (c+d x))}{d}+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {b C \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 b c C+4 A b d x+4 a B d x+2 b C d x+4 a A \text {arctanh}(\sin (c+d x))+4 (b B+a C) \sin (c+d x)+b C \sin (2 (c+d x))}{4 d} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \left (d x +c \right )+C \sin \left (d x +c \right ) a +A b \left (d x +c \right )+B \sin \left (d x +c \right ) b +C b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(82\) |
default | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B a \left (d x +c \right )+C \sin \left (d x +c \right ) a +A b \left (d x +c \right )+B \sin \left (d x +c \right ) b +C b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(82\) |
parallelrisch | \(\frac {-4 a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+C \sin \left (2 d x +2 c \right ) b +\left (4 B b +4 C a \right ) \sin \left (d x +c \right )+4 x d \left (B a +b \left (A +\frac {C}{2}\right )\right )}{4 d}\) | \(82\) |
parts | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A b +B a \right ) \left (d x +c \right )}{d}+\frac {\left (B b +C a \right ) \sin \left (d x +c \right )}{d}+\frac {C b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(83\) |
risch | \(x A b +x B a +\frac {b C x}{2}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B b}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C a}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B b}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C a}{2 d}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (2 d x +2 c \right ) C b}{4 d}\) | \(138\) |
norman | \(\frac {\left (A b +B a +\frac {1}{2} C b \right ) x +\left (A b +B a +\frac {1}{2} C b \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 A b +3 B a +\frac {3}{2} C b \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 A b +3 B a +\frac {3}{2} C b \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (2 B b +2 C a -C b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 B b +2 C a +C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 \left (B b +C a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(221\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {{\left (2 \, B a + {\left (2 \, A + C\right )} b\right )} d x + A a \log \left (\sin \left (d x + c\right ) + 1\right ) - A a \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (C b \cos \left (d x + c\right ) + 2 \, C a + 2 \, B b\right )} \sin \left (d x + c\right )}{2 \, d} \]
[In]
[Out]
\[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right ) \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} B a + 4 \, {\left (d x + c\right )} A b + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b + 4 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, C a \sin \left (d x + c\right ) + 4 \, B b \sin \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (65) = 130\).
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.30 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (2 \, B a + 2 \, A b + C b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 2.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.26 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {B\,b\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
[In]
[Out]