Integrand size = 29, antiderivative size = 61 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {(a A+2 b B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(A b+a B) \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3047, 3100, 2827, 3852, 8, 3855} \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {(a A+2 b B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(a B+A b) \tan (c+d x)}{d}+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d} \]
[In]
[Out]
Rule 8
Rule 2827
Rule 3047
Rule 3100
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {a A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (2 (A b+a B)+(a A+2 b B) \cos (c+d x)) \sec ^2(c+d x) \, dx \\ & = \frac {a A \sec (c+d x) \tan (c+d x)}{2 d}+(A b+a B) \int \sec ^2(c+d x) \, dx+\frac {1}{2} (a A+2 b B) \int \sec (c+d x) \, dx \\ & = \frac {(a A+2 b B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d}-\frac {(A b+a B) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {(a A+2 b B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(A b+a B) \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {a A \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b B \text {arctanh}(\sin (c+d x))}{d}+\frac {A b \tan (c+d x)}{d}+\frac {a B \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d} \]
[In]
[Out]
Time = 3.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {a A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B a \tan \left (d x +c \right )+A b \tan \left (d x +c \right )+B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(75\) |
default | \(\frac {a A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B a \tan \left (d x +c \right )+A b \tan \left (d x +c \right )+B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(75\) |
parts | \(\frac {a A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (A b +B a \right ) \tan \left (d x +c \right )}{d}+\frac {B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(76\) |
parallelrisch | \(\frac {-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1+\cos \left (2 d x +2 c \right )\right ) \left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (2 A b +2 B a \right ) \sin \left (2 d x +2 c \right )+2 a A \sin \left (d x +c \right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(114\) |
risch | \(-\frac {i \left (A a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 A b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 B a \,{\mathrm e}^{2 i \left (d x +c \right )}-a A \,{\mathrm e}^{i \left (d x +c \right )}-2 A b -2 B a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}\) | \(160\) |
norman | \(\frac {\frac {\left (a A -2 A b -2 B a \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (a A +2 A b +2 B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 a A -2 A b -2 B a \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (3 a A +2 A b +2 B a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {\left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(192\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.57 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {{\left (A a + 2 \, B b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A a + 2 \, B b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a + 2 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
[In]
[Out]
\[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=-\frac {A a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, B b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, B a \tan \left (d x + c\right ) - 4 \, A b \tan \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (57) = 114\).
Time = 0.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.48 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {{\left (A a + 2 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a + 2 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 1.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.70 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a+2\,A\,b+2\,B\,a\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,A\,b-A\,a+2\,B\,a\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A\,a+2\,B\,b\right )}{d} \]
[In]
[Out]