Integrand size = 39, antiderivative size = 17 \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\cos (a+b x) \sin (a+b x)}{b} \]
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Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {391} \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\sin (a+b x) \cos (a+b x)}{b} \]
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Rule 391
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-1+x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (a+b x)\right )}{b} \\ & = -\frac {\cos (a+b x) \sin (a+b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\cos (2 b x) \sin (2 a)}{2 b}-\frac {\cos (2 a) \sin (2 b x)}{2 b} \]
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Time = 0.74 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {\sin \left (2 x b +2 a \right )}{2 b}\) | \(15\) |
parallelrisch | \(-\frac {\sin \left (2 x b +2 a \right )}{2 b}\) | \(15\) |
derivativedivides | \(-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{b}\) | \(18\) |
default | \(-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{b}\) | \(18\) |
norman | \(\frac {\frac {2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b}-\frac {4 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{b}+\frac {2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{4}}\) | \(92\) |
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\cos \left (b x + a\right ) \sin \left (b x + a\right )}{b} \]
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\[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=- \int \frac {\csc ^{2}{\left (a + b x \right )}}{\csc ^{2}{\left (a + b x \right )} + \sec ^{2}{\left (a + b x \right )}}\, dx - \int \left (- \frac {\sec ^{2}{\left (a + b x \right )}}{\csc ^{2}{\left (a + b x \right )} + \sec ^{2}{\left (a + b x \right )}}\right )\, dx \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )} b} \]
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Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\sin \left (2 \, b x + 2 \, a\right )}{2 \, b} \]
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Time = 26.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx=-\frac {\sin \left (2\,a+2\,b\,x\right )}{2\,b} \]
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