Integrand size = 21, antiderivative size = 137 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^2(c+d x)}{2 a^3 d}-\frac {5 \sec ^3(c+d x)}{3 a^3 d}+\frac {5 \sec ^4(c+d x)}{4 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}-\frac {\sec ^6(c+d x)}{2 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d} \]
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Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3964, 90} \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\sec ^7(c+d x)}{7 a^3 d}-\frac {\sec ^6(c+d x)}{2 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}+\frac {5 \sec ^4(c+d x)}{4 a^3 d}-\frac {5 \sec ^3(c+d x)}{3 a^3 d}-\frac {\sec ^2(c+d x)}{2 a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}+\frac {\log (\cos (c+d x))}{a^3 d} \]
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Rule 90
Rule 3964
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(a-a x)^5 (a+a x)^2}{x^8} \, dx,x,\cos (c+d x)\right )}{a^{10} d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {a^7}{x^8}-\frac {3 a^7}{x^7}+\frac {a^7}{x^6}+\frac {5 a^7}{x^5}-\frac {5 a^7}{x^4}-\frac {a^7}{x^3}+\frac {3 a^7}{x^2}-\frac {a^7}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^{10} d} \\ & = \frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^2(c+d x)}{2 a^3 d}-\frac {5 \sec ^3(c+d x)}{3 a^3 d}+\frac {5 \sec ^4(c+d x)}{4 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}-\frac {\sec ^6(c+d x)}{2 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {(3732+4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))+2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))+630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (8+35 \log (\cos (c+d x)))) \sec ^7(c+d x)}{6720 a^3 d} \]
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Time = 4.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {\ln \left (\cos \left (d x +c \right )\right )+\frac {3}{\cos \left (d x +c \right )}+\frac {1}{5 \cos \left (d x +c \right )^{5}}-\frac {1}{2 \cos \left (d x +c \right )^{2}}+\frac {5}{4 \cos \left (d x +c \right )^{4}}-\frac {1}{2 \cos \left (d x +c \right )^{6}}+\frac {1}{7 \cos \left (d x +c \right )^{7}}-\frac {5}{3 \cos \left (d x +c \right )^{3}}}{d \,a^{3}}\) | \(86\) |
default | \(\frac {\ln \left (\cos \left (d x +c \right )\right )+\frac {3}{\cos \left (d x +c \right )}+\frac {1}{5 \cos \left (d x +c \right )^{5}}-\frac {1}{2 \cos \left (d x +c \right )^{2}}+\frac {5}{4 \cos \left (d x +c \right )^{4}}-\frac {1}{2 \cos \left (d x +c \right )^{6}}+\frac {1}{7 \cos \left (d x +c \right )^{7}}-\frac {5}{3 \cos \left (d x +c \right )^{3}}}{d \,a^{3}}\) | \(86\) |
risch | \(-\frac {i x}{a^{3}}-\frac {2 i c}{a^{3} d}+\frac {6 \,{\mathrm e}^{13 i \left (d x +c \right )}-2 \,{\mathrm e}^{12 i \left (d x +c \right )}+\frac {68 \,{\mathrm e}^{11 i \left (d x +c \right )}}{3}+10 \,{\mathrm e}^{10 i \left (d x +c \right )}+\frac {646 \,{\mathrm e}^{9 i \left (d x +c \right )}}{15}+8 \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {2488 \,{\mathrm e}^{7 i \left (d x +c \right )}}{35}+8 \,{\mathrm e}^{6 i \left (d x +c \right )}+\frac {646 \,{\mathrm e}^{5 i \left (d x +c \right )}}{15}+10 \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {68 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\) | \(203\) |
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Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.69 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {420 \, \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) + 1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{420 \, a^{3} d \cos \left (d x + c\right )^{7}} \]
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\[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\tan ^{11}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.66 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {420 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} + \frac {1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{a^{3} \cos \left (d x + c\right )^{7}}}{420 \, d} \]
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Time = 20.53 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.80 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {420 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac {420 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac {\frac {1393 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {819 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6755 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {20195 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {28749 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {8463 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 319}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \]
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Time = 18.40 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.64 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^3\,d}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {282\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {322\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {352}{105}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )} \]
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