Integrand size = 33, antiderivative size = 103 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=2 a b C x+\frac {\left (2 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 (A-2 C) \sin (c+d x)}{2 d}+\frac {a A b \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.36 (sec) , antiderivative size = 103, 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, 3110, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {\left (a^2 (A+2 C)+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a A b \tan (c+d x)}{d}+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+2 a b C x-\frac {b^2 (A-2 C) \sin (c+d x)}{2 d} \]
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Rule 2814
Rule 3102
Rule 3110
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+b \cos (c+d x)) \left (2 A b+a (A+2 C) \cos (c+d x)-b (A-2 C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a A b \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-2 A b^2-a^2 (A+2 C)-4 a b C \cos (c+d x)+b^2 (A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 (A-2 C) \sin (c+d x)}{2 d}+\frac {a A b \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-2 A b^2-a^2 (A+2 C)-4 a b C \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = 2 a b C x-\frac {b^2 (A-2 C) \sin (c+d x)}{2 d}+\frac {a A b \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \left (-2 A b^2-a^2 (A+2 C)\right ) \int \sec (c+d x) \, dx \\ & = 2 a b C x+\frac {\left (2 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 (A-2 C) \sin (c+d x)}{2 d}+\frac {a A b \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(103)=206\).
Time = 2.67 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.42 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {8 a b C (c+d x)-2 \left (2 A b^2+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (2 A b^2+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {8 a A b \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {a^2 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {8 a A b \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+4 b^2 C \sin (c+d x)}{4 d} \]
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Time = 6.45 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08
method | result | size |
parts | \(\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (A \,b^{2}+a^{2} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} C \sin \left (d x +c \right )}{d}+\frac {2 a A b \tan \left (d x +c \right )}{d}+\frac {2 C a b \left (d x +c \right )}{d}\) | \(111\) |
derivativedivides | \(\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 A a b \tan \left (d x +c \right )+2 C a b \left (d x +c \right )+A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{2}}{d}\) | \(112\) |
default | \(\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 A a b \tan \left (d x +c \right )+2 C a b \left (d x +c \right )+A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{2}}{d}\) | \(112\) |
parallelrisch | \(\frac {-\left (2 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (2 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+4 a b x d C \cos \left (2 d x +2 c \right )+4 A \sin \left (2 d x +2 c \right ) a b +C \sin \left (3 d x +3 c \right ) b^{2}+\left (2 A \,a^{2}+b^{2} C \right ) \sin \left (d x +c \right )+4 a b x d C}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(171\) |
risch | \(2 a b C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} b^{2} C}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} b^{2} C}{2 d}-\frac {i A a \left ({\mathrm e}^{3 i \left (d x +c \right )} a -4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a \,{\mathrm e}^{i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} C}{d}+\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} C}{d}\) | \(236\) |
norman | \(\frac {\frac {\left (A \,a^{2}-4 A a b +2 b^{2} C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (A \,a^{2}+4 A a b +2 b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (5 A \,a^{2}-12 A a b +2 b^{2} C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (5 A \,a^{2}+12 A a b +2 b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (5 A \,a^{2}-4 A a b -2 b^{2} C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (5 A \,a^{2}+4 A a b -2 b^{2} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+2 a b C x +4 a b C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a b C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a b C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a b C x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a b C x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a b C x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {\left (A \,a^{2}+2 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (A \,a^{2}+2 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(408\) |
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Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.35 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {8 \, C a b d x \cos \left (d x + c\right )^{2} + {\left ({\left (A + 2 \, C\right )} a^{2} + 2 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (A + 2 \, C\right )} a^{2} + 2 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{2} \cos \left (d x + c\right )^{2} + 4 \, A a b \cos \left (d x + c\right ) + A a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.36 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {8 \, {\left (d x + c\right )} C a b - A a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C b^{2} \sin \left (d x + c\right ) + 8 \, A a b \tan \left (d x + c\right )}{4 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.83 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} C a b + \frac {4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + {\left (A a^{2} + 2 \, C a^{2} + 2 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a^{2} + 2 \, C a^{2} + 2 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 2.44 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.83 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {2\,\left (\frac {A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+A\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,C\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\right )}{d}+\frac {\frac {C\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{2}+\frac {C\,b^2\,\sin \left (c+d\,x\right )}{4}+A\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \]
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