Integrand size = 15, antiderivative size = 17 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {\tan ^6(x)}{6}+\frac {\tan ^8(x)}{8} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4205, 2687, 14} \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {\tan ^8(x)}{8}+\frac {\tan ^6(x)}{6} \]
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Rule 14
Rule 2687
Rule 4205
Rubi steps \begin{align*} \text {integral}& = \int \sec ^4(x) \tan ^5(x) \, dx \\ & = \text {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (x)\right ) \\ & = \frac {\tan ^6(x)}{6}+\frac {\tan ^8(x)}{8} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {\sec ^4(x)}{4}-\frac {\sec ^6(x)}{3}+\frac {\sec ^8(x)}{8} \]
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Time = 19.94 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\tan \left (x \right )^{6}}{6}+\frac {\tan \left (x \right )^{8}}{8}\) | \(14\) |
default | \(\frac {\tan \left (x \right )^{6}}{6}+\frac {\tan \left (x \right )^{8}}{8}\) | \(14\) |
parts | \(\frac {\sec \left (x \right )^{8}}{8}+\frac {\sec \left (x \right )^{4}}{4}-\frac {\sec \left (x \right )^{6}}{3}\) | \(20\) |
risch | \(\frac {4 \,{\mathrm e}^{12 i x}-\frac {16 \,{\mathrm e}^{10 i x}}{3}+\frac {40 \,{\mathrm e}^{8 i x}}{3}-\frac {16 \,{\mathrm e}^{6 i x}}{3}+4 \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{8}}\) | \(48\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {6 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} + 3}{24 \, \cos \left (x\right )^{8}} \]
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Time = 0.91 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {\sec ^{8}{\left (x \right )}}{8} - \frac {\sec ^{6}{\left (x \right )}}{3} + \frac {\sec ^{4}{\left (x \right )}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (13) = 26\).
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.47 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {6 \, \sin \left (x\right )^{4} - 4 \, \sin \left (x\right )^{2} + 1}{24 \, {\left (\sin \left (x\right )^{8} - 4 \, \sin \left (x\right )^{6} + 6 \, \sin \left (x\right )^{4} - 4 \, \sin \left (x\right )^{2} + 1\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {6 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} + 3}{24 \, \cos \left (x\right )^{8}} \]
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Time = 26.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx=\frac {{\mathrm {tan}\left (x\right )}^6\,\left (3\,{\mathrm {tan}\left (x\right )}^2+4\right )}{24} \]
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