Integrand size = 31, antiderivative size = 104 \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{8 a^3 d}-\frac {4 i \sec ^3(c+d x)}{3 a^3 d}+\frac {i \sec ^5(c+d x)}{5 a^3 d}+\frac {7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d} \]
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Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3171, 3169, 3853, 3855, 2686, 30, 2691, 14} \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{8 a^3 d}+\frac {i \sec ^5(c+d x)}{5 a^3 d}-\frac {4 i \sec ^3(c+d x)}{3 a^3 d}-\frac {3 \tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac {7 \tan (c+d x) \sec (c+d x)}{8 a^3 d} \]
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Rule 14
Rule 30
Rule 2686
Rule 2691
Rule 3169
Rule 3171
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {i \int \sec ^6(c+d x) (i a \cos (c+d x)+a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {i \int \left (-i a^3 \sec ^3(c+d x)-3 a^3 \sec ^3(c+d x) \tan (c+d x)+3 i a^3 \sec ^3(c+d x) \tan ^2(c+d x)+a^3 \sec ^3(c+d x) \tan ^3(c+d x)\right ) \, dx}{a^6} \\ & = \frac {i \int \sec ^3(c+d x) \tan ^3(c+d x) \, dx}{a^3}-\frac {(3 i) \int \sec ^3(c+d x) \tan (c+d x) \, dx}{a^3}+\frac {\int \sec ^3(c+d x) \, dx}{a^3}-\frac {3 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{a^3} \\ & = \frac {\sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}+\frac {\int \sec (c+d x) \, dx}{2 a^3}+\frac {3 \int \sec ^3(c+d x) \, dx}{4 a^3}+\frac {i \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {(3 i) \text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{2 a^3 d}-\frac {i \sec ^3(c+d x)}{a^3 d}+\frac {7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d}+\frac {3 \int \sec (c+d x) \, dx}{8 a^3}+\frac {i \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {7 \text {arctanh}(\sin (c+d x))}{8 a^3 d}-\frac {4 i \sec ^3(c+d x)}{3 a^3 d}+\frac {i \sec ^5(c+d x)}{5 a^3 d}+\frac {7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {3 \sec ^3(c+d x) \tan (c+d x)}{4 a^3 d} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i \sec ^8(c+d x) (-i \cos (3 (c+d x))+\sin (3 (c+d x))) \left (448+1680 i \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right ) \cos ^5(c+d x)+640 \cos (2 (c+d x))-150 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))\right )}{960 a^3 d (-i+\tan (c+d x))^3} \]
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Time = 0.90 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.17
method | result | size |
risch | \(-\frac {i \left (105 \,{\mathrm e}^{9 i \left (d x +c \right )}+490 \,{\mathrm e}^{7 i \left (d x +c \right )}+896 \,{\mathrm e}^{5 i \left (d x +c \right )}+790 \,{\mathrm e}^{3 i \left (d x +c \right )}-105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{60 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {7 \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{8 d \,a^{3}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d \,a^{3}}\) | \(122\) |
derivativedivides | \(\frac {-\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {2 \left (\frac {1}{16}+\frac {13 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {2 \left (-\frac {3}{8}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2 \left (-\frac {5}{16}+\frac {11 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (-\frac {3}{4}+\frac {7 i}{24}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2 \left (\frac {5}{16}+\frac {11 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (\frac {3}{8}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2 \left (\frac {1}{16}-\frac {13 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (-\frac {3}{4}-\frac {7 i}{24}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{d \,a^{3}}\) | \(206\) |
default | \(\frac {-\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {2 \left (\frac {1}{16}+\frac {13 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {2 \left (-\frac {3}{8}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2 \left (-\frac {5}{16}+\frac {11 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (-\frac {3}{4}+\frac {7 i}{24}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2 \left (\frac {5}{16}+\frac {11 i}{16}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (\frac {3}{8}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2 \left (\frac {1}{16}-\frac {13 i}{16}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (-\frac {3}{4}-\frac {7 i}{24}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{d \,a^{3}}\) | \(206\) |
norman | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a d}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 a d}+\frac {34 i}{15 a d}-\frac {16 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a d}+\frac {6 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a d}-\frac {16 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 a d}+\frac {20 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d \,a^{3}}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \,a^{3}}\) | \(236\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (90) = 180\).
Time = 0.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.67 \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {105 \, {\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, {\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 980 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 1792 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 1580 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, e^{\left (i \, d x + i \, c\right )}}{120 \, {\left (a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
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\[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sec ^{6}{\left (c + d x \right )}}{- i \sin ^{3}{\left (c + d x \right )} - 3 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} + 3 i \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \cos ^{3}{\left (c + d x \right )}}\, dx}{a^{3}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (90) = 180\).
Time = 0.24 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.28 \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {\frac {16 \, {\left (-\frac {15 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {390 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {15 i \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 136\right )}}{-120 i \, a^{3} + \frac {600 i \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1200 i \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1200 i \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {600 i \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {120 i \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {7 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac {7 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{8 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.58 \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {\frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3}} - \frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{3}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 400 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 136 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a^{3}}}{120 \, d} \]
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Time = 25.99 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.44 \[ \int \frac {\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^3\,d}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,6{}\mathrm {i}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,16{}\mathrm {i}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,20{}\mathrm {i}}{3}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,16{}\mathrm {i}}{3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {34}{15}{}\mathrm {i}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^5} \]
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