Integrand size = 21, antiderivative size = 255 \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac {2 a \left (3 a^4-8 a^2 b^2+6 b^4\right ) \sec (c+d x)}{b^7 d}+\frac {\left (5 a^4-12 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}-\frac {4 a \left (a^2-2 b^2\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac {\left (3 a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}-\frac {2 a \sec ^5(c+d x)}{5 b^3 d}+\frac {\sec ^6(c+d x)}{6 b^2 d}+\frac {\left (a^2-b^2\right )^4}{a b^8 d (a+b \sec (c+d x))} \]
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Time = 0.25 (sec) , antiderivative size = 255, 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} \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (a^2-b^2\right )^4}{a b^8 d (a+b \sec (c+d x))}+\frac {\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac {4 a \left (a^2-2 b^2\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac {\left (3 a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}-\frac {\log (\cos (c+d x))}{a^2 d}-\frac {2 a \left (3 a^4-8 a^2 b^2+6 b^4\right ) \sec (c+d x)}{b^7 d}+\frac {\left (5 a^4-12 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}-\frac {2 a \sec ^5(c+d x)}{5 b^3 d}+\frac {\sec ^6(c+d x)}{6 b^2 d} \]
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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 )^4}{x (a+x)^2} \, dx,x,b \sec (c+d x)\right )}{b^8 d} \\ & = \frac {\text {Subst}\left (\int \left (-2 a \left (3 a^4-8 a^2 b^2+6 b^4\right )+\frac {b^8}{a^2 x}+\left (5 a^4-12 a^2 b^2+6 b^4\right ) x-4 a \left (a^2-2 b^2\right ) x^2+\left (3 a^2-4 b^2\right ) x^3-2 a x^4+x^5-\frac {\left (a^2-b^2\right )^4}{a (a+x)^2}+\frac {\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right )}{a^2 (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^8 d} \\ & = -\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac {2 a \left (3 a^4-8 a^2 b^2+6 b^4\right ) \sec (c+d x)}{b^7 d}+\frac {\left (5 a^4-12 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}-\frac {4 a \left (a^2-2 b^2\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac {\left (3 a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}-\frac {2 a \sec ^5(c+d x)}{5 b^3 d}+\frac {\sec ^6(c+d x)}{6 b^2 d}+\frac {\left (a^2-b^2\right )^4}{a b^8 d (a+b \sec (c+d x))} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.90 \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {-\frac {b^8 \log (\cos (c+d x))}{a^2}+\frac {\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2}-2 a b \left (3 a^4-8 a^2 b^2+6 b^4\right ) \sec (c+d x)+\frac {1}{2} b^2 \left (5 a^4-12 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)-\frac {4}{3} a b^3 \left (a^2-2 b^2\right ) \sec ^3(c+d x)+\frac {1}{4} b^4 \left (3 a^2-4 b^2\right ) \sec ^4(c+d x)-\frac {2}{5} a b^5 \sec ^5(c+d x)+\frac {1}{6} b^6 \sec ^6(c+d x)+\frac {\left (a^2-b^2\right )^4}{a (a+b \sec (c+d x))}}{b^8 d} \]
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Time = 3.60 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {-\frac {a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}{a^{2} b^{7} \left (b +a \cos \left (d x +c \right )\right )}+\frac {\left (7 a^{8}-20 a^{6} b^{2}+18 a^{4} b^{4}-4 a^{2} b^{6}-b^{8}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{8} a^{2}}-\frac {-3 a^{2}+4 b^{2}}{4 b^{4} \cos \left (d x +c \right )^{4}}-\frac {-5 a^{4}+12 a^{2} b^{2}-6 b^{4}}{2 b^{6} \cos \left (d x +c \right )^{2}}+\frac {\left (-7 a^{6}+20 a^{4} b^{2}-18 a^{2} b^{4}+4 b^{6}\right ) \ln \left (\cos \left (d x +c \right )\right )}{b^{8}}+\frac {1}{6 b^{2} \cos \left (d x +c \right )^{6}}-\frac {2 a}{5 b^{3} \cos \left (d x +c \right )^{5}}-\frac {4 a \left (a^{2}-2 b^{2}\right )}{3 b^{5} \cos \left (d x +c \right )^{3}}-\frac {2 a \left (3 a^{4}-8 a^{2} b^{2}+6 b^{4}\right )}{b^{7} \cos \left (d x +c \right )}}{d}\) | \(287\) |
default | \(\frac {-\frac {a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}{a^{2} b^{7} \left (b +a \cos \left (d x +c \right )\right )}+\frac {\left (7 a^{8}-20 a^{6} b^{2}+18 a^{4} b^{4}-4 a^{2} b^{6}-b^{8}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{8} a^{2}}-\frac {-3 a^{2}+4 b^{2}}{4 b^{4} \cos \left (d x +c \right )^{4}}-\frac {-5 a^{4}+12 a^{2} b^{2}-6 b^{4}}{2 b^{6} \cos \left (d x +c \right )^{2}}+\frac {\left (-7 a^{6}+20 a^{4} b^{2}-18 a^{2} b^{4}+4 b^{6}\right ) \ln \left (\cos \left (d x +c \right )\right )}{b^{8}}+\frac {1}{6 b^{2} \cos \left (d x +c \right )^{6}}-\frac {2 a}{5 b^{3} \cos \left (d x +c \right )^{5}}-\frac {4 a \left (a^{2}-2 b^{2}\right )}{3 b^{5} \cos \left (d x +c \right )^{3}}-\frac {2 a \left (3 a^{4}-8 a^{2} b^{2}+6 b^{4}\right )}{b^{7} \cos \left (d x +c \right )}}{d}\) | \(287\) |
risch | \(\text {Expression too large to display}\) | \(1191\) |
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Time = 0.37 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.66 \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {14 \, a^{3} b^{6} \cos \left (d x + c\right ) - 10 \, a^{2} b^{7} + 60 \, {\left (7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} + 30 \, {\left (7 \, a^{7} b^{2} - 20 \, a^{5} b^{4} + 18 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (7 \, a^{6} b^{3} - 20 \, a^{4} b^{5} + 18 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (7 \, a^{5} b^{4} - 20 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (7 \, a^{4} b^{5} - 20 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left ({\left (7 \, a^{9} - 20 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 4 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right )^{7} + {\left (7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7} - b^{9}\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + 60 \, {\left ({\left (7 \, a^{9} - 20 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 4 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{7} + {\left (7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (-\cos \left (d x + c\right )\right )}{60 \, {\left (a^{3} b^{8} d \cos \left (d x + c\right )^{7} + a^{2} b^{9} d \cos \left (d x + c\right )^{6}\right )}} \]
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\[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\tan ^{9}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.26 \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {14 \, a^{3} b^{5} \cos \left (d x + c\right ) - 10 \, a^{2} b^{6} + 60 \, {\left (7 \, a^{8} - 20 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{6} + 30 \, {\left (7 \, a^{7} b - 20 \, a^{5} b^{3} + 18 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (7 \, a^{6} b^{2} - 20 \, a^{4} b^{4} + 18 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (7 \, a^{5} b^{3} - 20 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (7 \, a^{4} b^{4} - 20 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2}}{a^{3} b^{7} \cos \left (d x + c\right )^{7} + a^{2} b^{8} \cos \left (d x + c\right )^{6}} + \frac {60 \, {\left (7 \, a^{6} - 20 \, a^{4} b^{2} + 18 \, a^{2} b^{4} - 4 \, b^{6}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{8}} - \frac {60 \, {\left (7 \, a^{8} - 20 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 4 \, a^{2} b^{6} - b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} b^{8}}}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1696 vs. \(2 (245) = 490\).
Time = 5.91 (sec) , antiderivative size = 1696, normalized size of antiderivative = 6.65 \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 16.98 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.98 \[ \int \frac {\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {2\,\left (-105\,a^7-105\,a^6\,b+265\,a^5\,b^2+265\,a^4\,b^3-191\,a^3\,b^4-191\,a^2\,b^5+15\,a\,b^6+15\,b^7\right )}{15\,a\,b^7}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (-42\,a^7-7\,a^6\,b+113\,a^5\,b^2+13\,a^4\,b^3-95\,a^3\,b^4-5\,a^2\,b^5+19\,a\,b^6+6\,b^7\right )}{a\,b^7}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (-210\,a^7-105\,a^6\,b+523\,a^5\,b^2+244\,a^4\,b^3-362\,a^3\,b^4-145\,a^2\,b^5+7\,a\,b^6+30\,b^7\right )}{3\,a\,b^7}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (-315\,a^7-105\,a^6\,b+809\,a^5\,b^2+223\,a^4\,b^3-613\,a^3\,b^4-99\,a^2\,b^5+91\,a\,b^6+45\,b^7\right )}{3\,a\,b^7}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-525\,a^7-350\,a^6\,b+1290\,a^5\,b^2+860\,a^4\,b^3-862\,a^3\,b^4-598\,a^2\,b^5+10\,a\,b^6+75\,b^7\right )}{5\,a\,b^7}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-630\,a^7-525\,a^6\,b+1555\,a^5\,b^2+1325\,a^4\,b^3-1067\,a^3\,b^4-955\,a^2\,b^5+45\,a\,b^6+90\,b^7\right )}{15\,a\,b^7}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (-7\,a^7+20\,a^5\,b^2-18\,a^3\,b^4+4\,a\,b^6+b^7\right )}{a\,b^7}}{d\,\left (\left (b-a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\left (7\,a-5\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (9\,b-21\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (35\,a-5\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (-35\,a-5\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (21\,a+9\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-7\,a-5\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a+b\right )}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (7\,a^6-20\,a^4\,b^2+18\,a^2\,b^4-4\,b^6\right )}{b^8\,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\,\left (7\,a^2+b^2\right )}{a^2\,b^8\,d} \]
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