Integrand size = 40, antiderivative size = 116 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\left (2 a b B+a^2 C+2 b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\left (2 a^2 B+3 b^2 B+6 a b C\right ) \tan (c+d x)}{3 d}+\frac {a (2 b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 B \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
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Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3108, 3067, 3100, 2827, 3852, 8, 3855} \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\left (a^2 C+2 a b B+2 b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\left (2 a^2 B+6 a b C+3 b^2 B\right ) \tan (c+d x)}{3 d}+\frac {a^2 B \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a (a C+2 b B) \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 8
Rule 2827
Rule 3067
Rule 3100
Rule 3108
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^4(c+d x) \, dx \\ & = \frac {a^2 B \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \int \left (-3 a (2 b B+a C)-\left (2 a^2 B+3 b^2 B+6 a b C\right ) \cos (c+d x)-3 b^2 C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {a (2 b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 B \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-2 \left (2 a^2 B+3 b^2 B+6 a b C\right )-3 \left (2 a b B+a^2 C+2 b^2 C\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a (2 b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 B \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \left (-2 a^2 B-3 b^2 B-6 a b C\right ) \int \sec ^2(c+d x) \, dx-\frac {1}{2} \left (-2 a b B-a^2 C-2 b^2 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {\left (2 a b B+a^2 C+2 b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a (2 b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 B \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {\left (2 a^2 B+3 b^2 B+6 a b C\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d} \\ & = \frac {\left (2 a b B+a^2 C+2 b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\left (2 a^2 B+3 b^2 B+6 a b C\right ) \tan (c+d x)}{3 d}+\frac {a (2 b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 B \sec ^2(c+d x) \tan (c+d x)}{3 d} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {3 \left (2 a b B+a^2 C+2 b^2 C\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (3 a (2 b B+a C) \sec (c+d x)+2 \left (3 a^2 B+3 b^2 B+6 a b C+a^2 B \tan ^2(c+d x)\right )\right )}{6 d} \]
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Time = 1.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02
method | result | size |
parts | \(\frac {\left (B \,b^{2}+2 C a b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (2 B a b +a^{2} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {B \,a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(118\) |
derivativedivides | \(\frac {a^{2} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+2 C a b \tan \left (d x +c \right )+2 B a b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+b^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{2} \tan \left (d x +c \right )}{d}\) | \(143\) |
default | \(\frac {a^{2} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+2 C a b \tan \left (d x +c \right )+2 B a b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+b^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{2} \tan \left (d x +c \right )}{d}\) | \(143\) |
parallelrisch | \(\frac {-9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (B a b +\frac {1}{2} a^{2} C +b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (B a b +\frac {1}{2} a^{2} C +b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (2 B \,a^{2}+3 B \,b^{2}+6 C a b \right ) \sin \left (3 d x +3 c \right )+3 \left (2 B a b +a^{2} C \right ) \sin \left (2 d x +2 c \right )+6 \left (B \,a^{2}+\frac {1}{2} B \,b^{2}+C a b \right ) \sin \left (d x +c \right )}{3 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(197\) |
risch | \(-\frac {i \left (6 B a b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 C \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-6 B \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 C a b \,{\mathrm e}^{4 i \left (d x +c \right )}-12 B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 B \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-24 C a b \,{\mathrm e}^{2 i \left (d x +c \right )}-6 B a b \,{\mathrm e}^{i \left (d x +c \right )}-3 C \,a^{2} {\mathrm e}^{i \left (d x +c \right )}-4 B \,a^{2}-6 B \,b^{2}-12 C a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} C}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2} C}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} C}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2} C}{d}\) | \(298\) |
norman | \(\frac {\frac {\left (2 B \,a^{2}+2 B a b -6 B \,b^{2}+a^{2} C -12 C a b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 B \,a^{2}+2 B a b +2 B \,b^{2}+a^{2} C +4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (2 B \,a^{2}-30 B a b +18 B \,b^{2}-15 a^{2} C +36 C a b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (2 B \,a^{2}-2 B a b -6 B \,b^{2}-a^{2} C -12 C a b \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 B \,a^{2}-2 B a b +2 B \,b^{2}-a^{2} C +4 C a b \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 B \,a^{2}+30 B a b +18 B \,b^{2}+15 a^{2} C +36 C a b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (14 B \,a^{2}-18 B a b +6 B \,b^{2}-9 a^{2} C +12 C a b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {\left (14 B \,a^{2}+18 B a b +6 B \,b^{2}+9 a^{2} C +12 C a b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {\left (2 B a b +a^{2} C +2 b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (2 B a b +a^{2} C +2 b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(456\) |
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Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {3 \, {\left (C a^{2} + 2 \, B a b + 2 \, C b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (C a^{2} + 2 \, B a b + 2 \, C b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, B a^{2} + 2 \, {\left (2 \, B a^{2} + 6 \, C a b + 3 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.48 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} - 3 \, C a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, B a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a b \tan \left (d x + c\right ) + 12 \, B b^{2} \tan \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (108) = 216\).
Time = 0.37 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.53 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {3 \, {\left (C a^{2} + 2 \, B a b + 2 \, C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (C a^{2} + 2 \, B a b + 2 \, C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 6.02 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.96 \[ \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {C\,a^2}{2}+B\,a\,b+C\,b^2\right )}{2\,C\,a^2+4\,B\,a\,b+4\,C\,b^2}\right )\,\left (C\,a^2+2\,B\,a\,b+2\,C\,b^2\right )}{d}-\frac {\left (2\,B\,a^2+2\,B\,b^2-C\,a^2-2\,B\,a\,b+4\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {4\,B\,a^2}{3}-8\,C\,a\,b-4\,B\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B\,a^2+2\,B\,b^2+C\,a^2+2\,B\,a\,b+4\,C\,a\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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