Integrand size = 40, antiderivative size = 80 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=b^2 C x+\frac {\left (a^2 B+2 b^2 B+4 a b C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a (2 b B+a C) \tan (c+d x)}{d}+\frac {a^2 B \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3108, 3067, 3100, 2814, 3855} \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {\left (a^2 B+4 a b C+2 b^2 B\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 B \tan (c+d x) \sec (c+d x)}{2 d}+\frac {a (a C+2 b B) \tan (c+d x)}{d}+b^2 C x \]
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Rule 2814
Rule 3067
Rule 3100
Rule 3108
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx \\ & = \frac {a^2 B \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-2 a (2 b B+a C)-\left (a^2 B+2 b^2 B+4 a b C\right ) \cos (c+d x)-2 b^2 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a (2 b B+a C) \tan (c+d x)}{d}+\frac {a^2 B \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^2 B-2 b^2 B-4 a b C-2 b^2 C \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = b^2 C x+\frac {a (2 b B+a C) \tan (c+d x)}{d}+\frac {a^2 B \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \left (-a^2 B-2 b^2 B-4 a b C\right ) \int \sec (c+d x) \, dx \\ & = b^2 C x+\frac {\left (a^2 B+2 b^2 B+4 a b C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a (2 b B+a C) \tan (c+d x)}{d}+\frac {a^2 B \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {2 b^2 C d x+\left (a^2 B+2 b^2 B+4 a b C\right ) \text {arctanh}(\sin (c+d x))+a (4 b B+2 a C+a B \sec (c+d x)) \tan (c+d x)}{2 d} \]
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Time = 1.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.30
method | result | size |
parts | \(\frac {\left (B \,b^{2}+2 C a b \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (2 B a b +a^{2} C \right ) \tan \left (d x +c \right )}{d}+\frac {B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{2} C \left (d x +c \right )}{d}\) | \(104\) |
derivativedivides | \(\frac {a^{2} C \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 B a b \tan \left (d x +c \right )+b^{2} C \left (d x +c \right )+B \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(112\) |
default | \(\frac {a^{2} C \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 B a b \tan \left (d x +c \right )+b^{2} C \left (d x +c \right )+B \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(112\) |
parallelrisch | \(\frac {-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (B \,a^{2}+2 B \,b^{2}+4 C a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1+\cos \left (2 d x +2 c \right )\right ) \left (B \,a^{2}+2 B \,b^{2}+4 C a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 C \,b^{2} d x \cos \left (2 d x +2 c \right )+\left (4 B a b +2 a^{2} C \right ) \sin \left (2 d x +2 c \right )+2 C \,b^{2} d x +2 B \sin \left (d x +c \right ) a^{2}}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(162\) |
risch | \(b^{2} C x -\frac {i a \left (B a \,{\mathrm e}^{3 i \left (d x +c \right )}-4 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 a C \,{\mathrm e}^{2 i \left (d x +c \right )}-B a \,{\mathrm e}^{i \left (d x +c \right )}-4 B b -2 a C \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C a b}{d}+\frac {B \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C a b}{d}\) | \(217\) |
norman | \(\frac {\frac {a \left (B a -4 B b -2 a C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (5 B a +4 B b +2 a C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+b^{2} C x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b^{2} C x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b^{2} C x +\frac {8 a \left (2 B b +a C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a \left (B a -2 B b -a C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a \left (B a +2 B b +a C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (B a +4 B b +2 a C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {a \left (5 B a -4 B b -2 a C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-b^{2} C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b^{2} C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b^{2} C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{2} C x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{2} C x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {\left (B \,a^{2}+2 B \,b^{2}+4 C a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (B \,a^{2}+2 B \,b^{2}+4 C a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(429\) |
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Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.70 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {4 \, C b^{2} d x \cos \left (d x + c\right )^{2} + {\left (B a^{2} + 4 \, C a b + 2 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (B a^{2} + 4 \, C a b + 2 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (B a^{2} + 2 \, {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.75 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} C b^{2} - B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C a^{2} \tan \left (d x + c\right ) + 8 \, B a b \tan \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (76) = 152\).
Time = 0.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.38 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {2 \, {\left (d x + c\right )} C b^{2} + {\left (B a^{2} + 4 \, C a b + 2 \, B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (B a^{2} + 4 \, C a b + 2 \, B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 2.53 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.20 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {2\,\left (\frac {B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+B\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+C\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,C\,a\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\right )}{d}+\frac {\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{2}+B\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \]
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