Integrand size = 39, antiderivative size = 101 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {(A b+a B+2 b C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(2 a A+3 b B+3 a C) \tan (c+d x)}{3 d}+\frac {(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a A \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
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Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3110, 3100, 2827, 3852, 8, 3855} \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {(a B+A b+2 b C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac {(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rule 8
Rule 2827
Rule 3100
Rule 3110
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \int \left (-3 (A b+a B)-(2 a A+3 b B+3 a C) \cos (c+d x)-3 b C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int (-2 (2 a A+3 b B+3 a C)-3 (A b+a B+2 b C) \cos (c+d x)) \sec ^2(c+d x) \, dx \\ & = \frac {(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} (-2 a A-3 b B-3 a C) \int \sec ^2(c+d x) \, dx-\frac {1}{2} (-A b-a B-2 b C) \int \sec (c+d x) \, dx \\ & = \frac {(A b+a B+2 b C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {(2 a A+3 b B+3 a C) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d} \\ & = \frac {(A b+a B+2 b C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(2 a A+3 b B+3 a C) \tan (c+d x)}{3 d}+\frac {(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a A \sec ^2(c+d x) \tan (c+d x)}{3 d} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 (A b+a B+2 b C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (6 b B+6 a (A+C)+3 (A b+a B) \sec (c+d x)+2 a A \tan ^2(c+d x)\right )}{6 d} \]
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Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.05
method | result | size |
parts | \(-\frac {a A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A b +B a \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (B b +C a \right ) \tan \left (d x +c \right )}{d}+\frac {C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(106\) |
derivativedivides | \(\frac {-a A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C a \tan \left (d x +c \right )+A b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B b \tan \left (d x +c \right )+C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(131\) |
default | \(\frac {-a A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B a \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C a \tan \left (d x +c \right )+A b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B b \tan \left (d x +c \right )+C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(131\) |
parallelrisch | \(\frac {-3 \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right ) \left (\left (A +2 C \right ) b +B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3 \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right ) \left (\left (A +2 C \right ) b +B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (6 B b +4 \left (A +\frac {3 C}{2}\right ) a \right ) \sin \left (3 d x +3 c \right )+\left (6 A b +6 B a \right ) \sin \left (2 d x +2 c \right )+12 \sin \left (d x +c \right ) \left (\frac {B b}{2}+a \left (A +\frac {C}{2}\right )\right )}{6 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(175\) |
risch | \(-\frac {i \left (3 A b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 B a \,{\mathrm e}^{5 i \left (d x +c \right )}-6 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 C a \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A a \,{\mathrm e}^{2 i \left (d x +c \right )}-12 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-12 C a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 A b \,{\mathrm e}^{i \left (d x +c \right )}-3 B a \,{\mathrm e}^{i \left (d x +c \right )}-4 a A -6 B b -6 C a \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C b}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A b}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C b}{d}\) | \(270\) |
norman | \(\frac {-\frac {2 \left (2 a A -A b -B a -2 B b -2 C a \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 a A -A b -B a +2 B b +2 C a \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (2 a A +A b +B a -2 B b -2 C a \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (2 a A +A b +B a +2 B b +2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {\left (14 a A -9 A b -9 B a +6 B b +6 C a \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (14 a A +9 A b +9 B a +6 B b +6 C a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {\left (A b +B a +2 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (A b +B a +2 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(306\) |
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Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.27 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 \, {\left (B a + {\left (A + 2 \, C\right )} b\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (B a + {\left (A + 2 \, C\right )} b\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left ({\left (2 \, A + 3 \, C\right )} a + 3 \, B b\right )} \cos \left (d x + c\right )^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right ) \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.60 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a - 3 \, B a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, A b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C a \tan \left (d x + c\right ) + 12 \, B b \tan \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (93) = 186\).
Time = 0.32 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.58 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 \, {\left (B a + A b + 2 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (B a + A b + 2 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 5.39 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.88 \[ \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A\,b}{2}+\frac {B\,a}{2}+C\,b\right )}{2\,A\,b+2\,B\,a+4\,C\,b}\right )\,\left (A\,b+B\,a+2\,C\,b\right )}{d}-\frac {\left (2\,A\,a-A\,b-B\,a+2\,B\,b+2\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {4\,A\,a}{3}-4\,B\,b-4\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a+A\,b+B\,a+2\,B\,b+2\,C\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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