Integrand size = 24, antiderivative size = 55 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d}+\frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \]
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Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d}-\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x) (a+x)^9 \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d}+\frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 (-i+\tan (c+d x))^{10} (6 i+5 \tan (c+d x))}{55 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (47 ) = 94\).
Time = 286.48 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25
method | result | size |
risch | \(\frac {1024 i a^{8} \left (55 \,{\mathrm e}^{18 i \left (d x +c \right )}+165 \,{\mathrm e}^{16 i \left (d x +c \right )}+330 \,{\mathrm e}^{14 i \left (d x +c \right )}+462 \,{\mathrm e}^{12 i \left (d x +c \right )}+462 \,{\mathrm e}^{10 i \left (d x +c \right )}+330 \,{\mathrm e}^{8 i \left (d x +c \right )}+165 \,{\mathrm e}^{6 i \left (d x +c \right )}+55 \,{\mathrm e}^{4 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{55 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{11}}\) | \(124\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+\frac {2 i a^{8}}{\cos \left (d x +c \right )^{4}}+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}\right )-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(339\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+\frac {2 i a^{8}}{\cos \left (d x +c \right )^{4}}+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}\right )-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(339\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (43) = 86\).
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.89 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {1024 \, {\left (-55 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 165 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 330 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 462 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 462 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 330 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 165 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 55 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 11 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{55 \, {\left (d e^{\left (22 i \, d x + 22 i \, c\right )} + 11 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 55 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 165 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 330 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 462 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 462 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 330 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 165 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 55 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
Time = 0.35 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.44 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
Time = 1.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.44 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \]
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Time = 4.73 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.95 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{10}+\frac {\sin \left (11\,c+11\,d\,x\right )}{110}+\frac {\cos \left (c+d\,x\right )\,63{}\mathrm {i}}{1280}+\frac {\cos \left (3\,c+3\,d\,x\right )\,9{}\mathrm {i}}{256}+\frac {\cos \left (5\,c+5\,d\,x\right )\,9{}\mathrm {i}}{512}+\frac {\cos \left (7\,c+7\,d\,x\right )\,3{}\mathrm {i}}{512}-\frac {\cos \left (9\,c+9\,d\,x\right )\,253{}\mathrm {i}}{2560}-\frac {\cos \left (11\,c+11\,d\,x\right )\,23{}\mathrm {i}}{2560}\right )}{d\,{\cos \left (c+d\,x\right )}^{11}} \]
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