Integrand size = 21, antiderivative size = 89 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {1}{16} (5 a+6 b) x+\frac {(5 a+6 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4130, 2715, 8} \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {(5 a+6 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {(5 a+6 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} x (5 a+6 b)+\frac {a \sin (e+f x) \cos ^5(e+f x)}{6 f} \]
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Rule 8
Rule 2715
Rule 4130
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{6} (5 a+6 b) \int \cos ^4(e+f x) \, dx \\ & = \frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{8} (5 a+6 b) \int \cos ^2(e+f x) \, dx \\ & = \frac {(5 a+6 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{16} (5 a+6 b) \int 1 \, dx \\ & = \frac {1}{16} (5 a+6 b) x+\frac {(5 a+6 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac {(5 a+6 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a \cos ^5(e+f x) \sin (e+f x)}{6 f} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {60 a e+72 b e+60 a f x+72 b f x+(45 a+48 b) \sin (2 (e+f x))+(9 a+6 b) \sin (4 (e+f x))+a \sin (6 (e+f x))}{192 f} \]
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Time = 0.50 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {\left (45 a +48 b \right ) \sin \left (2 f x +2 e \right )+\left (9 a +6 b \right ) \sin \left (4 f x +4 e \right )+a \sin \left (6 f x +6 e \right )+60 x f \left (a +\frac {6 b}{5}\right )}{192 f}\) | \(61\) |
risch | \(\frac {5 a x}{16}+\frac {3 x b}{8}+\frac {a \sin \left (6 f x +6 e \right )}{192 f}+\frac {3 \sin \left (4 f x +4 e \right ) a}{64 f}+\frac {\sin \left (4 f x +4 e \right ) b}{32 f}+\frac {15 a \sin \left (2 f x +2 e \right )}{64 f}+\frac {\sin \left (2 f x +2 e \right ) b}{4 f}\) | \(85\) |
derivativedivides | \(\frac {a \left (\frac {\left (\cos \left (f x +e \right )^{5}+\frac {5 \cos \left (f x +e \right )^{3}}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+b \left (\frac {\left (\cos \left (f x +e \right )^{3}+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(86\) |
default | \(\frac {a \left (\frac {\left (\cos \left (f x +e \right )^{5}+\frac {5 \cos \left (f x +e \right )^{3}}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+b \left (\frac {\left (\cos \left (f x +e \right )^{3}+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(86\) |
norman | \(\frac {\left (-\frac {5 a}{16}-\frac {3 b}{8}\right ) x +\left (-\frac {45 a}{16}-\frac {27 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-\frac {25 a}{16}-\frac {15 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {25 a}{16}-\frac {15 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (\frac {5 a}{16}+\frac {3 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}+\left (\frac {25 a}{16}+\frac {15 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (\frac {25 a}{16}+\frac {15 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}+\left (\frac {45 a}{16}+\frac {27 b}{8}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-\frac {\left (11 a +10 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {\left (11 a +10 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{8 f}+\frac {\left (15 a +2 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{2 f}+\frac {\left (19 a -6 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{12 f}-\frac {5 \left (19 a -6 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{24 f}-\frac {5 \left (19 a -6 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}+\frac {\left (19 a -6 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{12 f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\) | \(341\) |
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {3 \, {\left (5 \, a + 6 \, b\right )} f x + {\left (8 \, a \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a + 6 \, b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a + 6 \, b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f} \]
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Timed out. \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {3 \, {\left (f x + e\right )} {\left (5 \, a + 6 \, b\right )} + \frac {3 \, {\left (5 \, a + 6 \, b\right )} \tan \left (f x + e\right )^{5} + 8 \, {\left (5 \, a + 6 \, b\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (11 \, a + 10 \, b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1}}{48 \, f} \]
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Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {3 \, {\left (f x + e\right )} {\left (5 \, a + 6 \, b\right )} + \frac {15 \, a \tan \left (f x + e\right )^{5} + 18 \, b \tan \left (f x + e\right )^{5} + 40 \, a \tan \left (f x + e\right )^{3} + 48 \, b \tan \left (f x + e\right )^{3} + 33 \, a \tan \left (f x + e\right ) + 30 \, b \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3}}}{48 \, f} \]
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Time = 19.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=x\,\left (\frac {5\,a}{16}+\frac {3\,b}{8}\right )+\frac {\left (\frac {5\,a}{16}+\frac {3\,b}{8}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (\frac {5\,a}{6}+b\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (\frac {11\,a}{16}+\frac {5\,b}{8}\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^6+3\,{\mathrm {tan}\left (e+f\,x\right )}^4+3\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )} \]
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