Integrand size = 15, antiderivative size = 27 \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=-\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2686} \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=\frac {\sec ^3(a+b x)}{3 b}-\frac {\sec (a+b x)}{b} \]
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Rule 2686
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b} \\ & = -\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=-\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sec ^{3}\left (b x +a \right )\right )}{3}-\sec \left (b x +a \right )}{b}\) | \(24\) |
default | \(\frac {\frac {\left (\sec ^{3}\left (b x +a \right )\right )}{3}-\sec \left (b x +a \right )}{b}\) | \(24\) |
norman | \(\frac {-\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {4}{3 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3}}\) | \(39\) |
parallelrisch | \(\frac {\frac {4}{3}-4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}\) | \(47\) |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{5 i \left (b x +a \right )}+2 \,{\mathrm e}^{3 i \left (b x +a \right )}+3 \,{\mathrm e}^{i \left (b x +a \right )}\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{3}}\) | \(53\) |
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Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=-\frac {3 \, \cos \left (b x + a\right )^{2} - 1}{3 \, b \cos \left (b x + a\right )^{3}} \]
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Timed out. \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=-\frac {3 \, \cos \left (b x + a\right )^{2} - 1}{3 \, b \cos \left (b x + a\right )^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=-\frac {3 \, \cos \left (b x + a\right )^{2} - 1}{3 \, b \cos \left (b x + a\right )^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \sec (a+b x) \tan ^3(a+b x) \, dx=-\frac {{\cos \left (a+b\,x\right )}^2-\frac {1}{3}}{b\,{\cos \left (a+b\,x\right )}^3} \]
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