Integrand size = 24, antiderivative size = 55 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d}+\frac {i (a+i a \tan (c+d x))^8}{8 a^3 d} \]
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Time = 0.05 (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} \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {i (a+i a \tan (c+d x))^8}{8 a^3 d}-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x) (a+x)^6 \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {Subst}\left (\int \left (2 a (a+x)^6-(a+x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d}+\frac {i (a+i a \tan (c+d x))^8}{8 a^3 d} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {i a^5 (-i+\tan (c+d x))^7 (9 i+7 \tan (c+d x))}{56 d} \]
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Time = 66.86 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {32 i a^{5} \left (28 \,{\mathrm e}^{12 i \left (d x +c \right )}+56 \,{\mathrm e}^{10 i \left (d x +c \right )}+70 \,{\mathrm e}^{8 i \left (d x +c \right )}+56 \,{\mathrm e}^{6 i \left (d x +c \right )}+28 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{7 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\) | \(91\) |
derivativedivides | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{4 \cos \left (d x +c \right )^{4}}-a^{5} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(213\) |
default | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{4 \cos \left (d x +c \right )^{4}}-a^{5} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(213\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (43) = 86\).
Time = 0.24 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.47 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {32 \, {\left (-28 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 56 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 70 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 56 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 28 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{5}\right )}}{7 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=i a^{5} \left (\int \left (- i \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (43) = 86\).
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.96 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {-7 i \, a^{5} \tan \left (d x + c\right )^{8} - 40 \, a^{5} \tan \left (d x + c\right )^{7} + 84 i \, a^{5} \tan \left (d x + c\right )^{6} + 56 \, a^{5} \tan \left (d x + c\right )^{5} + 70 i \, a^{5} \tan \left (d x + c\right )^{4} + 168 \, a^{5} \tan \left (d x + c\right )^{3} - 140 i \, a^{5} \tan \left (d x + c\right )^{2} - 56 \, a^{5} \tan \left (d x + c\right )}{56 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (43) = 86\).
Time = 0.78 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.96 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {-7 i \, a^{5} \tan \left (d x + c\right )^{8} - 40 \, a^{5} \tan \left (d x + c\right )^{7} + 84 i \, a^{5} \tan \left (d x + c\right )^{6} + 56 \, a^{5} \tan \left (d x + c\right )^{5} + 70 i \, a^{5} \tan \left (d x + c\right )^{4} + 168 \, a^{5} \tan \left (d x + c\right )^{3} - 140 i \, a^{5} \tan \left (d x + c\right )^{2} - 56 \, a^{5} \tan \left (d x + c\right )}{56 \, d} \]
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Time = 3.89 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.75 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5\,\sin \left (c+d\,x\right )\,\left (56\,{\cos \left (c+d\,x\right )}^7+{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )\,140{}\mathrm {i}-168\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3\,70{}\mathrm {i}-56\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^5\,84{}\mathrm {i}+40\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^6+{\sin \left (c+d\,x\right )}^7\,7{}\mathrm {i}\right )}{56\,d\,{\cos \left (c+d\,x\right )}^8} \]
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