Integrand size = 33, antiderivative size = 148 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \]
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Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3098, 2829, 2729, 2727} \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rule 2727
Rule 2729
Rule 2829
Rule 3098
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {\int \frac {-a (3 A+4 B-4 C)-7 a C \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = \frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(6 A+8 B+13 C) \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2} \\ & = \frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {(6 A+8 B+13 C) \int \frac {1}{a+a \cos (c+d x)} \, dx}{105 a^3} \\ & = \frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Time = 1.72 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.41 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (70 (3 A+2 B+4 C) \sin \left (\frac {d x}{2}\right )-35 (4 B+5 C) \sin \left (c+\frac {d x}{2}\right )+126 A \sin \left (c+\frac {3 d x}{2}\right )+168 B \sin \left (c+\frac {3 d x}{2}\right )+168 C \sin \left (c+\frac {3 d x}{2}\right )-105 C \sin \left (2 c+\frac {3 d x}{2}\right )+42 A \sin \left (2 c+\frac {5 d x}{2}\right )+56 B \sin \left (2 c+\frac {5 d x}{2}\right )+91 C \sin \left (2 c+\frac {5 d x}{2}\right )+6 A \sin \left (3 c+\frac {7 d x}{2}\right )+8 B \sin \left (3 c+\frac {7 d x}{2}\right )+13 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{420 a^4 d (1+\cos (c+d x))^4} \]
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Time = 1.75 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (A -B +C \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 \left (3 A -C -B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+7 \left (A +\frac {B}{3}-\frac {C}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 A +7 B +7 C \right )}{56 a^{4} d}\) | \(90\) |
derivativedivides | \(\frac {\frac {\left (A -B +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -C -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A +B -C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{8 d \,a^{4}}\) | \(106\) |
default | \(\frac {\frac {\left (A -B +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -C -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A +B -C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{8 d \,a^{4}}\) | \(106\) |
risch | \(\frac {2 i \left (105 C \,{\mathrm e}^{5 i \left (d x +c \right )}+140 B \,{\mathrm e}^{4 i \left (d x +c \right )}+175 C \,{\mathrm e}^{4 i \left (d x +c \right )}+210 A \,{\mathrm e}^{3 i \left (d x +c \right )}+140 B \,{\mathrm e}^{3 i \left (d x +c \right )}+280 C \,{\mathrm e}^{3 i \left (d x +c \right )}+126 A \,{\mathrm e}^{2 i \left (d x +c \right )}+168 B \,{\mathrm e}^{2 i \left (d x +c \right )}+168 C \,{\mathrm e}^{2 i \left (d x +c \right )}+42 A \,{\mathrm e}^{i \left (d x +c \right )}+56 B \,{\mathrm e}^{i \left (d x +c \right )}+91 C \,{\mathrm e}^{i \left (d x +c \right )}+6 A +8 B +13 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(177\) |
norman | \(\frac {\frac {\left (A -B +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (A +B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (9 A +7 B +5 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}+\frac {\left (27 A +11 B +C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}+\frac {\left (31 A -17 B +3 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}+\frac {\left (123 A -11 B -31 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{420 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{3}}\) | \(181\) |
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Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\left ({\left (6 \, A + 8 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (6 \, A + 8 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (39 \, A + 52 \, B + 32 \, C\right )} \cos \left (d x + c\right ) + 36 \, A + 13 \, B + 8 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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Time = 2.10 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.78 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} - \frac {B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right )}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.75 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {B {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.16 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
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Time = 1.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (B-3\,A+C\right )}{40\,a^4\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+B+C\right )}{8\,a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A+B-C\right )}{24\,a^4\,d} \]
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