Integrand size = 24, antiderivative size = 94 \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 i a^2 \sec ^3(c+d x)}{12 d}+\frac {5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d} \]
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Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3579, 3567, 3853, 3855} \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 i a^2 \sec ^3(c+d x)}{12 d}+\frac {i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {5 a^2 \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rule 3567
Rule 3579
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {1}{4} (5 a) \int \sec ^3(c+d x) (a+i a \tan (c+d x)) \, dx \\ & = \frac {5 i a^2 \sec ^3(c+d x)}{12 d}+\frac {i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {1}{4} \left (5 a^2\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {5 i a^2 \sec ^3(c+d x)}{12 d}+\frac {5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {1}{8} \left (5 a^2\right ) \int \sec (c+d x) \, dx \\ & = \frac {5 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 i a^2 \sec ^3(c+d x)}{12 d}+\frac {5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.89 \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {2 i a^2 \sec ^3(c+d x)}{3 d}+\frac {5 a^2 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {a^2 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 3.99 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18
method | result | size |
risch | \(-\frac {i a^{2} \left (15 \,{\mathrm e}^{7 i \left (d x +c \right )}-73 \,{\mathrm e}^{5 i \left (d x +c \right )}-55 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(111\) |
derivativedivides | \(\frac {-a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {2 i a^{2}}{3 \cos \left (d x +c \right )^{3}}+a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(121\) |
default | \(\frac {-a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {2 i a^{2}}{3 \cos \left (d x +c \right )^{3}}+a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(121\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (80) = 160\).
Time = 0.25 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.72 \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {-30 i \, a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 146 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 110 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 30 i \, a^{2} e^{\left (i \, d x + i \, c\right )} + 15 \, {\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \, {\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{24 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=- a^{2} \left (\int \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \left (- 2 i \tan {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\right )\, dx + \int \left (- \sec ^{3}{\left (c + d x \right )}\right )\, dx\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.38 \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {3 \, a^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {32 i \, a^{2}}{\cos \left (d x + c\right )^{3}}}{48 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (80) = 160\).
Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.84 \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {15 \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 15 \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) + \frac {2 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 33 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 33 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 i \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 7.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.11 \[ \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {5\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,4{}\mathrm {i}+\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,4{}\mathrm {i}+\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,4{}\mathrm {i}}{3}-\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {a^2\,4{}\mathrm {i}}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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