Integrand size = 24, antiderivative size = 84 \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {3 \text {arctanh}(\sin (c+d x))}{8 a d}-\frac {i \sec ^5(c+d x)}{5 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a d} \]
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Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3582, 3853, 3855} \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {3 \text {arctanh}(\sin (c+d x))}{8 a d}-\frac {i \sec ^5(c+d x)}{5 a d}+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac {3 \tan (c+d x) \sec (c+d x)}{8 a d} \]
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Rule 3582
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {i \sec ^5(c+d x)}{5 a d}+\frac {\int \sec ^5(c+d x) \, dx}{a} \\ & = -\frac {i \sec ^5(c+d x)}{5 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {3 \int \sec ^3(c+d x) \, dx}{4 a} \\ & = -\frac {i \sec ^5(c+d x)}{5 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {3 \int \sec (c+d x) \, dx}{8 a} \\ & = \frac {3 \text {arctanh}(\sin (c+d x))}{8 a d}-\frac {i \sec ^5(c+d x)}{5 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a d} \\ \end{align*}
Time = 0.82 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {240 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right )+\sec ^5(c+d x) (-64 i+70 \sin (2 (c+d x))+15 \sin (4 (c+d x)))}{320 a d} \]
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Time = 0.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.45
method | result | size |
risch | \(-\frac {i \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}+70 \,{\mathrm e}^{7 i \left (d x +c \right )}+128 \,{\mathrm e}^{5 i \left (d x +c \right )}-70 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{20 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 a d}\) | \(122\) |
derivativedivides | \(\frac {\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {2 \left (\frac {7}{16}+\frac {5 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (\frac {5}{16}+\frac {3 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {2 \left (\frac {1}{4}+\frac {3 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (\frac {1}{8}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}-\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2 \left (\frac {5}{16}-\frac {3 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (\frac {1}{4}-\frac {3 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (-\frac {1}{8}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2 \left (-\frac {7}{16}+\frac {5 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{a d}\) | \(206\) |
default | \(\frac {\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {2 \left (\frac {7}{16}+\frac {5 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (\frac {5}{16}+\frac {3 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {2 \left (\frac {1}{4}+\frac {3 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (\frac {1}{8}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}-\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2 \left (\frac {5}{16}-\frac {3 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (\frac {1}{4}-\frac {3 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (-\frac {1}{8}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2 \left (-\frac {7}{16}+\frac {5 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{a d}\) | \(206\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (74) = 148\).
Time = 0.25 (sec) , antiderivative size = 266, normalized size of antiderivative = 3.17 \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {15 \, {\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \, {\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 30 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 140 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 256 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 140 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 30 i \, e^{\left (i \, d x + i \, c\right )}}{40 \, {\left (a d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
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\[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\sec ^{7}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.43 \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {3 \, {\left (\frac {16 \, {\left (\frac {25 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {80 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {40 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {25 i \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 8\right )}}{-120 i \, a + \frac {600 i \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1200 i \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1200 i \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {600 i \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {120 i \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}\right )}}{8 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.64 \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\frac {15 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a} - \frac {15 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a} + \frac {2 \, {\left (25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 40 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 80 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a}}{40 \, d} \]
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Time = 8.36 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.30 \[ \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a\,d}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2\,a}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2\,a}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4\,a}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,4{}\mathrm {i}}{a}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,2{}\mathrm {i}}{a}+\frac {2{}\mathrm {i}}{5\,a}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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