Integrand size = 19, antiderivative size = 107 \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=2 a b \left (2 a^2+b^2\right ) x+\frac {a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d} \]
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2872, 3112, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\frac {a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+2 a b x \left (2 a^2+b^2\right )+\frac {4 a b^3 \sin (c+d x) \cos (c+d x)}{3 d}+\frac {b^2 \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
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Rule 2814
Rule 2872
Rule 3102
Rule 3112
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x)) \left (3 a^3+b \left (9 a^2+2 b^2\right ) \cos (c+d x)+8 a b^2 \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^4+12 a b \left (2 a^2+b^2\right ) \cos (c+d x)+2 b^2 \left (17 a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^4+12 a b \left (2 a^2+b^2\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = 2 a b \left (2 a^2+b^2\right ) x+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+a^4 \int \sec (c+d x) \, dx \\ & = 2 a b \left (2 a^2+b^2\right ) x+\frac {a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.20 \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\frac {24 a b \left (2 a^2+b^2\right ) (c+d x)-12 a^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 b^2 \left (8 a^2+b^2\right ) \sin (c+d x)+12 a b^3 \sin (2 (c+d x))+b^4 \sin (3 (c+d x))}{12 d} \]
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Time = 2.62 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{3} b \left (d x +c \right )+6 \sin \left (d x +c \right ) a^{2} b^{2}+4 a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(98\) |
default | \(\frac {a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{3} b \left (d x +c \right )+6 \sin \left (d x +c \right ) a^{2} b^{2}+4 a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(98\) |
parallelrisch | \(\frac {-12 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+12 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+12 \sin \left (2 d x +2 c \right ) a \,b^{3}+\sin \left (3 d x +3 c \right ) b^{4}+9 \left (8 a^{2} b^{2}+b^{4}\right ) \sin \left (d x +c \right )+48 b d \left (a^{2}+\frac {b^{2}}{2}\right ) x a}{12 d}\) | \(104\) |
parts | \(\frac {a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {4 a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {6 \sin \left (d x +c \right ) a^{2} b^{2}}{d}+\frac {4 a^{3} b \left (d x +c \right )}{d}\) | \(109\) |
risch | \(4 a^{3} b x +2 a \,b^{3} x -\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b^{2}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b^{2}}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (3 d x +3 c \right ) b^{4}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a \,b^{3}}{d}\) | \(169\) |
norman | \(\frac {\left (4 a^{3} b +2 a \,b^{3}\right ) x +\left (4 a^{3} b +2 a \,b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (16 a^{3} b +8 a \,b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (16 a^{3} b +8 a \,b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (24 a^{3} b +12 a \,b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 b^{2} \left (6 a^{2}-2 a b +b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 b^{2} \left (6 a^{2}+2 a b +b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 b^{2} \left (54 a^{2}-6 a b +5 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 b^{2} \left (54 a^{2}+6 a b +5 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(307\) |
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\frac {3 \, a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 12 \, {\left (2 \, a^{3} b + a b^{3}\right )} d x + 2 \, {\left (b^{4} \cos \left (d x + c\right )^{2} + 6 \, a b^{3} \cos \left (d x + c\right ) + 18 \, a^{2} b^{2} + 2 \, b^{4}\right )} \sin \left (d x + c\right )}{6 \, d} \]
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\[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{4} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89 \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\frac {12 \, {\left (d x + c\right )} a^{3} b + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} - {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{4} + 3 \, a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 18 \, a^{2} b^{2} \sin \left (d x + c\right )}{3 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (101) = 202\).
Time = 0.32 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.98 \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\frac {3 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 6 \, {\left (2 \, a^{3} b + a b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
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Time = 14.75 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.48 \[ \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx=\frac {3\,b^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {6\,a^2\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {4\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,a^3\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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