Integrand size = 21, antiderivative size = 103 \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {4 \tan (c+d x)}{a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {4 \tan ^3(c+d x)}{3 a d} \]
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Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2847, 2827, 3852, 3853, 3855} \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {4 \tan ^3(c+d x)}{3 a d}+\frac {4 \tan (c+d x)}{a d}-\frac {3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac {\tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
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Rule 2827
Rule 2847
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {\int (-4 a+3 a \cos (c+d x)) \sec ^4(c+d x) \, dx}{a^2} \\ & = -\frac {\sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {3 \int \sec ^3(c+d x) \, dx}{a}+\frac {4 \int \sec ^4(c+d x) \, dx}{a} \\ & = -\frac {3 \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {3 \int \sec (c+d x) \, dx}{2 a}-\frac {4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d} \\ & = -\frac {3 \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {4 \tan (c+d x)}{a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {\sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {4 \tan ^3(c+d x)}{3 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(368\) vs. \(2(103)=206\).
Time = 3.13 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.57 \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (6 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+\frac {1}{8} \cos \left (\frac {1}{2} (c+d x)\right ) \sec (c) \sec ^3(c+d x) \left (9 \cos (2 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 \cos (4 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+27 \cos (d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+27 \cos (2 c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-9 \cos (2 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 \cos (4 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 \sin (d x)-12 \sin (2 c+d x)-6 \sin (c+2 d x)-6 \sin (3 c+2 d x)+20 \sin (2 c+3 d x)\right )\right )}{3 a d (1+\cos (c+d x))} \]
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Time = 1.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.28
method | result | size |
norman | \(\frac {\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\) | \(132\) |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}}{d a}\) | \(134\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}}{d a}\) | \(134\) |
parallelrisch | \(\frac {\left (27 \cos \left (d x +c \right )+9 \cos \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-27 \cos \left (d x +c \right )-9 \cos \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+44 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {7 \cos \left (2 d x +2 c \right )}{22}+\frac {4 \cos \left (3 d x +3 c \right )}{11}+\frac {1}{2}\right )}{6 a d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(137\) |
risch | \(\frac {i \left (9 \,{\mathrm e}^{6 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+24 \,{\mathrm e}^{4 i \left (d x +c \right )}+24 \,{\mathrm e}^{3 i \left (d x +c \right )}+39 \,{\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}+16\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d a}\) | \(147\) |
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.20 \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {9 \, {\left (\cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, {\left (\cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, \cos \left (d x + c\right )^{3} + 7 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (97) = 194\).
Time = 0.21 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.99 \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a - \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{6 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \]
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Time = 14.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93 \[ \int \frac {\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d}-\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^3} \]
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