Integrand size = 33, antiderivative size = 109 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{2} \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) x+\frac {2 a A b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b (A-C) \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^2 \tan (c+d x)}{d} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3127, 3112, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{2} x \left (C \left (2 a^2+b^2\right )+2 A b^2\right )+\frac {2 a A b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b (A-C) \sin (c+d x)}{d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac {b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d} \]
[In]
[Out]
Rule 2814
Rule 3102
Rule 3112
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\int (a+b \cos (c+d x)) \left (2 A b+a C \cos (c+d x)-b (2 A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int \left (4 a A b+\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \cos (c+d x)-4 a b (A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {2 a b (A-C) \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int \left (4 a A b+\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) x-\frac {2 a b (A-C) \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^2 \tan (c+d x)}{d}+(2 a A b) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) x+\frac {2 a A b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b (A-C) \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^2 \tan (c+d x)}{d} \\ \end{align*}
Time = 2.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.21 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {2 \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) (c+d x)-8 a A b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 a A b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 a b C \sin (c+d x)+\left (4 a^2 A+b^2 C+b^2 C \cos (2 (c+d x))\right ) \tan (c+d x)}{4 d} \]
[In]
[Out]
Time = 4.54 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {A \,a^{2} \tan \left (d x +c \right )+a^{2} C \left (d x +c \right )+2 A a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 a b \sin \left (d x +c \right ) C +A \,b^{2} \left (d x +c \right )+b^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(94\) |
default | \(\frac {A \,a^{2} \tan \left (d x +c \right )+a^{2} C \left (d x +c \right )+2 A a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 a b \sin \left (d x +c \right ) C +A \,b^{2} \left (d x +c \right )+b^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(94\) |
parts | \(\frac {a^{2} A \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{2}+a^{2} C \right ) \left (d x +c \right )}{d}+\frac {b^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 \sin \left (d x +c \right ) C a b}{d}\) | \(102\) |
parallelrisch | \(\frac {-16 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a b \cos \left (d x +c \right )+16 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a b \cos \left (d x +c \right )+8 a b \sin \left (2 d x +2 c \right ) C +C \sin \left (3 d x +3 c \right ) b^{2}+8 x d \left (\left (A +\frac {C}{2}\right ) b^{2}+a^{2} C \right ) \cos \left (d x +c \right )+\sin \left (d x +c \right ) \left (8 A \,a^{2}+b^{2} C \right )}{8 d \cos \left (d x +c \right )}\) | \(133\) |
risch | \(x A \,b^{2}+a^{2} C x +\frac {b^{2} C x}{2}-\frac {i b^{2} C \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i C a b \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {i C a b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i b^{2} C \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i A \,a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 A a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 A a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(160\) |
norman | \(\frac {\left (-A \,b^{2}-a^{2} C -\frac {1}{2} b^{2} C \right ) x +\left (-3 A \,b^{2}-3 a^{2} C -\frac {3}{2} b^{2} C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (A \,b^{2}+a^{2} C +\frac {1}{2} b^{2} C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 A \,b^{2}+3 a^{2} C +\frac {3}{2} b^{2} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 A \,b^{2}-2 a^{2} C -b^{2} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 A \,b^{2}+2 a^{2} C +b^{2} C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (6 A \,a^{2}-b^{2} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 A \,a^{2}-4 C a b +b^{2} C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 A \,a^{2}+4 C a b +b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {8 a \left (a A -C b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a \left (a A +C b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 A a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 A a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(394\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {2 \, A a b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, A a b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, C a^{2} + {\left (2 \, A + C\right )} b^{2}\right )} d x \cos \left (d x + c\right ) + {\left (C b^{2} \cos \left (d x + c\right )^{2} + 4 \, C a b \cos \left (d x + c\right ) + 2 \, A a^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
[In]
[Out]
\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec ^{2}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} C a^{2} + 4 \, {\left (d x + c\right )} A b^{2} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 4 \, A a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, C a b \sin \left (d x + c\right ) + 4 \, A a^{2} \tan \left (d x + c\right )}{4 \, d} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.61 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {4 \, A a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 4 \, A a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + {\left (2 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (4 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 1.93 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.77 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {2\,C\,a\,b\,\sin \left (c+d\,x\right )}{d}+\frac {C\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}-\frac {A\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{d}-\frac {A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,4{}\mathrm {i}}{d} \]
[In]
[Out]