Integrand size = 17, antiderivative size = 46 \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {\sec ^7(a+b x)}{7 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^{11}(a+b x)}{11 b} \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2686, 276} \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {\sec ^{11}(a+b x)}{11 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^7(a+b x)}{7 b} \]
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Rule 276
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\sec (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sec (a+b x)\right )}{b} \\ & = \frac {\sec ^7(a+b x)}{7 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^{11}(a+b x)}{11 b} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {\sec ^7(a+b x)}{7 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^{11}(a+b x)}{11 b} \]
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Time = 0.49 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sec ^{11}\left (b x +a \right )\right )}{11}-\frac {2 \left (\sec ^{9}\left (b x +a \right )\right )}{9}+\frac {\left (\sec ^{7}\left (b x +a \right )\right )}{7}}{b}\) | \(36\) |
default | \(\frac {\frac {\left (\sec ^{11}\left (b x +a \right )\right )}{11}-\frac {2 \left (\sec ^{9}\left (b x +a \right )\right )}{9}+\frac {\left (\sec ^{7}\left (b x +a \right )\right )}{7}}{b}\) | \(36\) |
risch | \(\frac {\frac {128 \,{\mathrm e}^{15 i \left (b x +a \right )}}{7}-\frac {2560 \,{\mathrm e}^{13 i \left (b x +a \right )}}{63}+\frac {47360 \,{\mathrm e}^{11 i \left (b x +a \right )}}{693}-\frac {2560 \,{\mathrm e}^{9 i \left (b x +a \right )}}{63}+\frac {128 \,{\mathrm e}^{7 i \left (b x +a \right )}}{7}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{11}}\) | \(75\) |
parallelrisch | \(\frac {-\frac {16}{693}-\frac {32 \left (\tan ^{16}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}-\frac {80 \left (\tan ^{14}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}-\frac {176 \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}-48 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {240 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{7}-\frac {80 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{63}+\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{63}}{b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{11}}\) | \(127\) |
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Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {99 \, \cos \left (b x + a\right )^{4} - 154 \, \cos \left (b x + a\right )^{2} + 63}{693 \, b \cos \left (b x + a\right )^{11}} \]
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Timed out. \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {99 \, \cos \left (b x + a\right )^{4} - 154 \, \cos \left (b x + a\right )^{2} + 63}{693 \, b \cos \left (b x + a\right )^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (40) = 80\).
Time = 0.36 (sec) , antiderivative size = 204, normalized size of antiderivative = 4.43 \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {16 \, {\left (\frac {11 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {55 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {297 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {2079 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {2541 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac {1155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac {462 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + 1\right )}}{693 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{11}} \]
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Time = 0.77 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx=\frac {99\,{\cos \left (a+b\,x\right )}^4-154\,{\cos \left (a+b\,x\right )}^2+63}{693\,b\,{\cos \left (a+b\,x\right )}^{11}} \]
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