Integrand size = 9, antiderivative size = 22 \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3 x}{2}+\frac {3 \tan (x)}{2}-\frac {1}{2} \sin ^2(x) \tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {294, 327, 209} \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3 x}{2}+\frac {3 \tan (x)}{2}-\frac {1}{2} \sin ^2(x) \tan (x) \]
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Rule 209
Rule 294
Rule 327
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right ) \\ & = -\frac {1}{2} \sin ^2(x) \tan (x)+\frac {3}{2} \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {3 \tan (x)}{2}-\frac {1}{2} \sin ^2(x) \tan (x)-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = -\frac {3 x}{2}+\frac {3 \tan (x)}{2}-\frac {1}{2} \sin ^2(x) \tan (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3 x}{2}+\frac {1}{4} \sin (2 x)+\tan (x) \]
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Time = 0.98 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {\cos \left (x \right ) \sin \left (x \right )}{2}-\frac {3 x}{2}+\tan \left (x \right )\) | \(13\) |
parallelrisch | \(-\frac {3 x}{2}+\frac {\sin \left (2 x \right )}{4}+\tan \left (x \right )\) | \(13\) |
parts | \(\frac {\cos \left (x \right ) \sin \left (x \right )}{2}-\frac {3 x}{2}+\tan \left (x \right )\) | \(13\) |
risch | \(-\frac {3 x}{2}-\frac {i {\mathrm e}^{2 i x}}{8}+\frac {i {\mathrm e}^{-2 i x}}{8}+\frac {2 i}{{\mathrm e}^{2 i x}+1}\) | \(33\) |
norman | \(\frac {\frac {3 x}{2}-2 \tan \left (\frac {x}{2}\right )^{3}-3 \tan \left (\frac {x}{2}\right )^{5}+\frac {3 x \tan \left (\frac {x}{2}\right )^{2}}{2}-\frac {3 x \tan \left (\frac {x}{2}\right )^{4}}{2}-\frac {3 x \tan \left (\frac {x}{2}\right )^{6}}{2}-3 \tan \left (\frac {x}{2}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\) | \(75\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3 \, x \cos \left (x\right ) - {\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{2 \, \cos \left (x\right )} \]
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Time = 0.58 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int (-\cos (x)+\sec (x))^2 \, dx=- \frac {3 x}{2} + \frac {\sin {\left (2 x \right )}}{4} + \tan {\left (x \right )} \]
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Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55 \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3}{2} \, x + \frac {1}{4} \, \sin \left (2 \, x\right ) + \tan \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3}{2} \, x + \frac {\tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} + \tan \left (x\right ) \]
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Time = 26.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int (-\cos (x)+\sec (x))^2 \, dx=-\frac {3\,x}{2}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+3\,\mathrm {tan}\left (\frac {x}{2}\right )}{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right )\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2} \]
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