Integrand size = 31, antiderivative size = 76 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{2 a^3 d}-\frac {4 i \sec (c+d x)}{a^3 d}+\frac {i \sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^3 d} \]
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Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3171, 3169, 3855, 2686, 8, 2691} \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{2 a^3 d}+\frac {i \sec ^3(c+d x)}{3 a^3 d}-\frac {4 i \sec (c+d x)}{a^3 d}-\frac {3 \tan (c+d x) \sec (c+d x)}{2 a^3 d} \]
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Rule 8
Rule 2686
Rule 2691
Rule 3169
Rule 3171
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {i \int \sec ^4(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 (c+d x)-3 a^3 \sec (c+d x) \tan (c+d x)+3 i a^3 \sec (c+d x) \tan ^2(c+d x)+a^3 \sec (c+d x) \tan ^3(c+d x)\right ) \, dx}{a^6} \\ & = \frac {i \int \sec (c+d x) \tan ^3(c+d x) \, dx}{a^3}-\frac {(3 i) \int \sec (c+d x) \tan (c+d x) \, dx}{a^3}+\frac {\int \sec (c+d x) \, dx}{a^3}-\frac {3 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{a^3} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^3 d}+\frac {3 \int \sec (c+d x) \, dx}{2 a^3}+\frac {i \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {(3 i) \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{a^3 d} \\ & = \frac {5 \text {arctanh}(\sin (c+d x))}{2 a^3 d}-\frac {4 i \sec (c+d x)}{a^3 d}+\frac {i \sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec (c+d x) \tan (c+d x)}{2 a^3 d} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {i \left (-60 i \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right )+\sec ^3(c+d x) (-20-24 \cos (2 (c+d x))+9 i \sin (2 (c+d x)))\right )}{12 a^3 d} \]
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Time = 0.80 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {i \left (15 \,{\mathrm e}^{5 i \left (d x +c \right )}+40 \,{\mathrm e}^{3 i \left (d x +c \right )}+33 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d \,a^{3}}+\frac {5 \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d \,a^{3}}\) | \(100\) |
derivativedivides | \(\frac {-\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (-\frac {3}{4}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (-\frac {3}{4}+\frac {7 i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (\frac {3}{4}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (-\frac {3}{4}-\frac {7 i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}}{d \,a^{3}}\) | \(138\) |
default | \(\frac {-\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (-\frac {3}{4}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (-\frac {3}{4}+\frac {7 i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (\frac {3}{4}-\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (-\frac {3}{4}-\frac {7 i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}}{d \,a^{3}}\) | \(138\) |
norman | \(\frac {-\frac {16 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}+\frac {22 i}{3 a d}+\frac {6 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{3}}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{3}}\) | \(158\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (66) = 132\).
Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.39 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {15 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 30 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 80 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 66 i \, e^{\left (i \, d x + i \, c\right )}}{6 \, {\left (a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
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\[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{- i \sin ^{3}{\left (c + d x \right )} - 3 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} + 3 i \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \cos ^{3}{\left (c + d x \right )}}\, dx}{a^{3}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (66) = 132\).
Time = 0.23 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.83 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {\frac {4 \, {\left (-\frac {9 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {9 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 22\right )}}{6 i \, a^{3} - \frac {18 i \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 i \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6 i \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {5 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac {5 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{2 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {\frac {15 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3}} - \frac {15 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{3}} - \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 22 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{3}}}{6 \, d} \]
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Time = 24.93 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.78 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx=\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}+\frac {\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,16{}\mathrm {i}}{a^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,6{}\mathrm {i}}{a^3}+\frac {22{}\mathrm {i}}{3\,a^3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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