Integrand size = 21, antiderivative size = 170 \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\log (\cos (c+d x))}{a d}-\frac {\left (a^2-b^2\right )^3 \log (a+b \sec (c+d x))}{a b^6 d}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \sec (c+d x)}{b^5 d}-\frac {a \left (a^2-3 b^2\right ) \sec ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2-3 b^2\right ) \sec ^3(c+d x)}{3 b^3 d}-\frac {a \sec ^4(c+d x)}{4 b^2 d}+\frac {\sec ^5(c+d x)}{5 b d} \]
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Time = 0.16 (sec) , antiderivative size = 170, 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 = {3970, 908} \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^3 \log (a+b \sec (c+d x))}{a b^6 d}-\frac {a \left (a^2-3 b^2\right ) \sec ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2-3 b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \sec (c+d x)}{b^5 d}-\frac {a \sec ^4(c+d x)}{4 b^2 d}+\frac {\log (\cos (c+d x))}{a d}+\frac {\sec ^5(c+d x)}{5 b d} \]
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Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^3}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{b^6 d} \\ & = -\frac {\text {Subst}\left (\int \left (-a^4 \left (1+\frac {3 b^2 \left (-a^2+b^2\right )}{a^4}\right )+\frac {b^6}{a x}+a \left (a^2-3 b^2\right ) x-\left (a^2-3 b^2\right ) x^2+a x^3-x^4+\frac {\left (a^2-b^2\right )^3}{a (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^6 d} \\ & = \frac {\log (\cos (c+d x))}{a d}-\frac {\left (a^2-b^2\right )^3 \log (a+b \sec (c+d x))}{a b^6 d}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \sec (c+d x)}{b^5 d}-\frac {a \left (a^2-3 b^2\right ) \sec ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2-3 b^2\right ) \sec ^3(c+d x)}{3 b^3 d}-\frac {a \sec ^4(c+d x)}{4 b^2 d}+\frac {\sec ^5(c+d x)}{5 b d} \\ \end{align*}
Time = 6.23 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {-\frac {b^6 \log (\cos (c+d x))}{a}+\frac {\left (a^2-b^2\right )^3 \log (a+b \sec (c+d x))}{a}-b \left (a^4-3 a^2 b^2+3 b^4\right ) \sec (c+d x)+\frac {1}{2} a b^2 \left (a^2-3 b^2\right ) \sec ^2(c+d x)-\frac {1}{3} b^3 \left (a^2-3 b^2\right ) \sec ^3(c+d x)+\frac {1}{4} a b^4 \sec ^4(c+d x)-\frac {1}{5} b^5 \sec ^5(c+d x)}{b^6 d} \]
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Time = 1.68 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {-\frac {a}{4 b^{2} \cos \left (d x +c \right )^{4}}-\frac {-a^{2}+3 b^{2}}{3 b^{3} \cos \left (d x +c \right )^{3}}-\frac {-a^{4}+3 a^{2} b^{2}-3 b^{4}}{b^{5} \cos \left (d x +c \right )}-\frac {\left (a^{2}-3 b^{2}\right ) a}{2 b^{4} \cos \left (d x +c \right )^{2}}+\frac {\left (a^{4}-3 a^{2} b^{2}+3 b^{4}\right ) a \ln \left (\cos \left (d x +c \right )\right )}{b^{6}}+\frac {1}{5 b \cos \left (d x +c \right )^{5}}+\frac {\left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{6} a}}{d}\) | \(184\) |
default | \(\frac {-\frac {a}{4 b^{2} \cos \left (d x +c \right )^{4}}-\frac {-a^{2}+3 b^{2}}{3 b^{3} \cos \left (d x +c \right )^{3}}-\frac {-a^{4}+3 a^{2} b^{2}-3 b^{4}}{b^{5} \cos \left (d x +c \right )}-\frac {\left (a^{2}-3 b^{2}\right ) a}{2 b^{4} \cos \left (d x +c \right )^{2}}+\frac {\left (a^{4}-3 a^{2} b^{2}+3 b^{4}\right ) a \ln \left (\cos \left (d x +c \right )\right )}{b^{6}}+\frac {1}{5 b \cos \left (d x +c \right )^{5}}+\frac {\left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{6} a}}{d}\) | \(184\) |
risch | \(-\frac {i x}{a}-\frac {2 i c}{a d}+\frac {2 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-6 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-2 a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+6 a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+8 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}-\frac {64 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}}{3}+16 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-6 a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+14 a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-\frac {92 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}}{3}+\frac {132 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{5}-6 a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+14 a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+8 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-\frac {64 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{3}+16 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-2 a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{i \left (d x +c \right )}-6 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{i \left (d x +c \right )}}{d \,b^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{6} d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{4} d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}-\frac {a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{6} d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{4} d}-\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a d}\) | \(598\) |
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Time = 0.34 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {15 \, a^{2} b^{4} \cos \left (d x + c\right ) + 60 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{5} \log \left (a \cos \left (d x + c\right ) + b\right ) - 60 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\cos \left (d x + c\right )\right ) - 12 \, a b^{5} - 60 \, {\left (a^{5} b - 3 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} + 30 \, {\left (a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 20 \, {\left (a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}}{60 \, a b^{6} d \cos \left (d x + c\right )^{5}} \]
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\[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\tan ^{7}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08 \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {60 \, {\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{6}} - \frac {60 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{6}} - \frac {15 \, a b^{3} \cos \left (d x + c\right ) - 60 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{4} - 12 \, b^{4} + 30 \, {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - 20 \, {\left (a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2}}{b^{5} \cos \left (d x + c\right )^{5}}}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (162) = 324\).
Time = 3.41 (sec) , antiderivative size = 1052, normalized size of antiderivative = 6.19 \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]
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Time = 15.60 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.32 \[ \int \frac {\tan ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (a^4-3\,a^2\,b^2+3\,b^4\right )}{b^6\,d}-\frac {\frac {2\,\left (15\,a^4-40\,a^2\,b^2+33\,b^4\right )}{15\,b^5}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (a^4+a^3\,b-2\,a^2\,b^2-2\,a\,b^3+b^4\right )}{b^5}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^4+3\,a^3\,b-10\,a^2\,b^2-8\,a\,b^3+6\,b^4\right )}{b^5}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (12\,a^4+3\,a^3\,b-34\,a^2\,b^2-6\,a\,b^3+30\,b^4\right )}{3\,b^5}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (18\,a^4+9\,a^3\,b-50\,a^2\,b^2-24\,a\,b^3+48\,b^4\right )}{3\,b^5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,{\left (a^2-b^2\right )}^3}{a\,b^6\,d} \]
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