Integrand size = 17, antiderivative size = 31 \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b} \]
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Time = 0.02 (sec) , antiderivative size = 31, 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, 14} \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=\frac {\sec ^5(a+b x)}{5 b}-\frac {\sec ^3(a+b x)}{3 b} \]
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Rule 14
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (a+b x)\right )}{b} \\ & = -\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b} \]
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Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sec ^{5}\left (b x +a \right )\right )}{5}-\frac {\left (\sec ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(26\) |
default | \(\frac {\frac {\left (\sec ^{5}\left (b x +a \right )\right )}{5}-\frac {\left (\sec ^{3}\left (b x +a \right )\right )}{3}}{b}\) | \(26\) |
risch | \(-\frac {8 \left (5 \,{\mathrm e}^{7 i \left (b x +a \right )}-2 \,{\mathrm e}^{5 i \left (b x +a \right )}+5 \,{\mathrm e}^{3 i \left (b x +a \right )}\right )}{15 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{5}}\) | \(53\) |
norman | \(\frac {\frac {4}{15 b}-\frac {4 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{5}}\) | \(71\) |
parallelrisch | \(\frac {\frac {4}{15}-4 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}-\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3}}{b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{5} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{5}}\) | \(73\) |
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {5 \, \cos \left (b x + a\right )^{2} - 3}{15 \, b \cos \left (b x + a\right )^{5}} \]
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Timed out. \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {5 \, \cos \left (b x + a\right )^{2} - 3}{15 \, b \cos \left (b x + a\right )^{5}} \]
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Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {5 \, \cos \left (b x + a\right )^{2} - 3}{15 \, b \cos \left (b x + a\right )^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \sec ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {\frac {{\cos \left (a+b\,x\right )}^2}{3}-\frac {1}{5}}{b\,{\cos \left (a+b\,x\right )}^5} \]
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