Integrand size = 15, antiderivative size = 41 \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {\sec (a+b x)}{b}-\frac {2 \sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b} \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2686, 200} \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {\sec ^5(a+b x)}{5 b}-\frac {2 \sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b} \]
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Rule 200
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (a+b x)\right )}{b} \\ & = \frac {\sec (a+b x)}{b}-\frac {2 \sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {\sec (a+b x)}{b}-\frac {2 \sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b} \]
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Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sec ^{5}\left (b x +a \right )\right )}{5}-\frac {2 \left (\sec ^{3}\left (b x +a \right )\right )}{3}+\sec \left (b x +a \right )}{b}\) | \(32\) |
default | \(\frac {\frac {\left (\sec ^{5}\left (b x +a \right )\right )}{5}-\frac {2 \left (\sec ^{3}\left (b x +a \right )\right )}{3}+\sec \left (b x +a \right )}{b}\) | \(32\) |
norman | \(\frac {-\frac {16}{15 b}+\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {32 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{5}}\) | \(55\) |
parallelrisch | \(\frac {-\frac {16}{15}-\frac {32 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}+\frac {16 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}}{b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{5} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{5}}\) | \(60\) |
risch | \(\frac {2 \,{\mathrm e}^{9 i \left (b x +a \right )}+\frac {8 \,{\mathrm e}^{7 i \left (b x +a \right )}}{3}+\frac {116 \,{\mathrm e}^{5 i \left (b x +a \right )}}{15}+\frac {8 \,{\mathrm e}^{3 i \left (b x +a \right )}}{3}+2 \,{\mathrm e}^{i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{5}}\) | \(75\) |
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {15 \, \cos \left (b x + a\right )^{4} - 10 \, \cos \left (b x + a\right )^{2} + 3}{15 \, b \cos \left (b x + a\right )^{5}} \]
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Timed out. \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {15 \, \cos \left (b x + a\right )^{4} - 10 \, \cos \left (b x + a\right )^{2} + 3}{15 \, b \cos \left (b x + a\right )^{5}} \]
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Time = 0.36 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.76 \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {16 \, {\left (\frac {5 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}}{15 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \sec (a+b x) \tan ^5(a+b x) \, dx=\frac {15\,{\cos \left (a+b\,x\right )}^4-10\,{\cos \left (a+b\,x\right )}^2+3}{15\,b\,{\cos \left (a+b\,x\right )}^5} \]
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