Integrand size = 13, antiderivative size = 32 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\frac {1}{4} \text {arctanh}(2 \cos (x) \sin (x))+\frac {\sec ^2(x) \tan (x)}{2 \left (1-\tan ^2(x)\right )^2} \]
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Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {424, 21, 212} \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\frac {1}{4} \text {arctanh}(2 \sin (x) \cos (x))+\frac {\tan (x) \sec ^2(x)}{2 \left (1-\tan ^2(x)\right )^2} \]
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Rule 21
Rule 212
Rule 424
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\tan (x)\right ) \\ & = \frac {\sec ^2(x) \tan (x)}{2 \left (1-\tan ^2(x)\right )^2}-\frac {1}{4} \text {Subst}\left (\int \frac {-2+2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tan (x)\right ) \\ & = \frac {\sec ^2(x) \tan (x)}{2 \left (1-\tan ^2(x)\right )^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{4} \text {arctanh}(2 \cos (x) \sin (x))+\frac {\sec ^2(x) \tan (x)}{2 \left (1-\tan ^2(x)\right )^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\frac {1}{4} \text {arctanh}(\sin (2 x))+\frac {1}{4} \sec (2 x) \tan (2 x) \]
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Time = 0.77 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50
method | result | size |
default | \(-\frac {1}{4 \left (1+\tan \left (x \right )\right )^{2}}+\frac {1}{4+4 \tan \left (x \right )}+\frac {\ln \left (1+\tan \left (x \right )\right )}{4}+\frac {1}{4 \left (\tan \left (x \right )-1\right )^{2}}+\frac {1}{4 \tan \left (x \right )-4}-\frac {\ln \left (\tan \left (x \right )-1\right )}{4}\) | \(48\) |
risch | \(-\frac {i \left ({\mathrm e}^{6 i x}-{\mathrm e}^{2 i x}\right )}{2 \left ({\mathrm e}^{4 i x}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}-i\right )}{4}+\frac {\ln \left (i+{\mathrm e}^{2 i x}\right )}{4}\) | \(49\) |
parallelrisch | \(\frac {\left (1+\cos \left (4 x \right )\right ) \ln \left (\frac {-\cos \left (x \right )-\sin \left (x \right )}{\cos \left (x \right )+1}\right )+\left (-\cos \left (4 x \right )-1\right ) \ln \left (\frac {-\cos \left (x \right )+\sin \left (x \right )}{\cos \left (x \right )+1}\right )+2 \sin \left (2 x \right )}{4 \cos \left (4 x \right )+4}\) | \(67\) |
norman | \(\frac {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )^{5}-\tan \left (\frac {x}{2}\right )^{7}+\tan \left (\frac {x}{2}\right )}{\left (\tan \left (\frac {x}{2}\right )^{4}-6 \tan \left (\frac {x}{2}\right )^{2}+1\right )^{2}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )-1\right )}{4}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )^{2}+2 \tan \left (\frac {x}{2}\right )-1\right )}{4}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\frac {{\left (4 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 1\right )} \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - {\left (4 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) + 4 \, \cos \left (x\right ) \sin \left (x\right )}{8 \, {\left (4 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (29) = 58\).
Time = 1.61 (sec) , antiderivative size = 765, normalized size of antiderivative = 23.91 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\text {Too large to display} \]
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none
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\frac {\tan \left (x\right )^{3} + \tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 1\right )}} + \frac {1}{4} \, \log \left (\tan \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (\tan \left (x\right ) - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=-\frac {\sin \left (2 \, x\right )}{4 \, {\left (\sin \left (2 \, x\right )^{2} - 1\right )}} + \frac {1}{8} \, \log \left (\sin \left (2 \, x\right ) + 1\right ) - \frac {1}{8} \, \log \left (-\sin \left (2 \, x\right ) + 1\right ) \]
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Time = 27.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^3} \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (x\right )\right )}{2}+\frac {\frac {{\mathrm {tan}\left (x\right )}^3}{2}+\frac {\mathrm {tan}\left (x\right )}{2}}{{\mathrm {tan}\left (x\right )}^4-2\,{\mathrm {tan}\left (x\right )}^2+1} \]
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