Integrand size = 21, antiderivative size = 108 \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=4 a b^3 x+\frac {a^2 \left (a^2+12 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2871, 3110, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 \left (a^2+12 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+4 a b^3 x \]
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Rule 2814
Rule 2871
Rule 3102
Rule 3110
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+b \cos (c+d x)) \left (6 a^2 b+a \left (a^2+6 b^2\right ) \cos (c+d x)-b \left (a^2-2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^2 \left (a^2+12 b^2\right )-8 a b^3 \cos (c+d x)+b^2 \left (a^2-2 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^2 \left (a^2+12 b^2\right )-8 a b^3 \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = 4 a b^3 x-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2 \left (a^2+12 b^2\right )\right ) \int \sec (c+d x) \, dx \\ & = 4 a b^3 x+\frac {a^2 \left (a^2+12 b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 2.00 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.61 \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\frac {a \left (16 b^3 c+16 b^3 d x-2 a \left (a^2+12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a \left (a^2+12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )+4 b^4 \sin (c+d x)+16 a^3 b \tan (c+d x)}{4 d} \]
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Time = 3.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b \tan \left (d x +c \right )+6 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a \,b^{3} \left (d x +c \right )+\sin \left (d x +c \right ) b^{4}}{d}\) | \(96\) |
default | \(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b \tan \left (d x +c \right )+6 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a \,b^{3} \left (d x +c \right )+\sin \left (d x +c \right ) b^{4}}{d}\) | \(96\) |
parts | \(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) b^{4}}{d}+\frac {4 a^{3} b \tan \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a \,b^{3} \left (d x +c \right )}{d}\) | \(107\) |
parallelrisch | \(\frac {-a^{2} \left (a^{2}+12 b^{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+a^{2} \left (a^{2}+12 b^{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+8 a \,b^{3} d x \cos \left (2 d x +2 c \right )+8 \sin \left (2 d x +2 c \right ) a^{3} b +\sin \left (3 d x +3 c \right ) b^{4}+\left (2 a^{4}+b^{4}\right ) \sin \left (d x +c \right )+8 a \,b^{3} d x}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(162\) |
risch | \(4 a \,b^{3} x -\frac {i {\mathrm e}^{i \left (d x +c \right )} b^{4}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} b^{4}}{2 d}-\frac {i a^{3} \left (a \,{\mathrm e}^{3 i \left (d x +c \right )}-8 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a \,{\mathrm e}^{i \left (d x +c \right )}-8 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{d}\) | \(196\) |
norman | \(\frac {\frac {\left (a^{4}-8 a^{3} b +2 b^{4}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (a^{4}+8 a^{3} b +2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (5 a^{4}-24 a^{3} b +2 b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (5 a^{4}+24 a^{3} b +2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+4 a \,b^{3} x +\frac {2 \left (5 a^{4}-8 a^{3} b -2 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (5 a^{4}+8 a^{3} b -2 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+8 a \,b^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \,b^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \,b^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 a \,b^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a \,b^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a^{2} \left (a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{2} \left (a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(395\) |
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Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.20 \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\frac {16 \, a b^{3} d x \cos \left (d x + c\right )^{2} + {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{3} b \cos \left (d x + c\right ) + a^{4}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{4} \sec ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\frac {16 \, {\left (d x + c\right )} a b^{3} - a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, b^{4} \sin \left (d x + c\right ) + 16 \, a^{3} b \tan \left (d x + c\right )}{4 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.64 \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\frac {8 \, {\left (d x + c\right )} a b^{3} + \frac {4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
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Time = 14.53 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.41 \[ \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx=\frac {b^4\,\sin \left (c+d\,x\right )}{d}+\frac {a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {12\,a^2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,a^3\,b\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
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