Integrand size = 39, antiderivative size = 72 \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=\frac {\log \left (1-\sqrt {2} \tan (a+b x)+\tan ^2(a+b x)\right )}{2 \sqrt {2} b}-\frac {\log \left (1+\sqrt {2} \tan (a+b x)+\tan ^2(a+b x)\right )}{2 \sqrt {2} b} \]
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Time = 1.59 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1179, 642} \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=\frac {\log \left (\tan ^2(a+b x)-\sqrt {2} \tan (a+b x)+1\right )}{2 \sqrt {2} b}-\frac {\log \left (\tan ^2(a+b x)+\sqrt {2} \tan (a+b x)+1\right )}{2 \sqrt {2} b} \]
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Rule 642
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-1+x^2}{1+x^4} \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\tan (a+b x)\right )}{2 \sqrt {2} b}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\tan (a+b x)\right )}{2 \sqrt {2} b} \\ & = \frac {\log \left (1-\sqrt {2} \tan (a+b x)+\tan ^2(a+b x)\right )}{2 \sqrt {2} b}-\frac {\log \left (1+\sqrt {2} \tan (a+b x)+\tan ^2(a+b x)\right )}{2 \sqrt {2} b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.36 \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=-\frac {\text {arctanh}\left (\frac {\sin (2 a+2 b x)}{\sqrt {2}}\right )}{\sqrt {2} b} \]
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Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{4 i \left (x b +a \right )}-2 i \sqrt {2}\, {\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{4 b}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{4 i \left (x b +a \right )}+2 i \sqrt {2}\, {\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{4 b}\) | \(72\) |
derivativedivides | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}{1-\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )+1\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )-1\right )\right )}{8}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}{1+\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )+1\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )-1\right )\right )}{8}}{b}\) | \(168\) |
default | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}{1-\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )+1\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )-1\right )\right )}{8}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}{1+\sqrt {2}\, \tan \left (x b +a \right )+\tan \left (x b +a \right )^{2}}\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )+1\right )+2 \arctan \left (\sqrt {2}\, \tan \left (x b +a \right )-1\right )\right )}{8}}{b}\) | \(168\) |
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Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03 \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=\frac {\sqrt {2} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{4} + 2 \, \sqrt {2} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, \cos \left (b x + a\right )^{2} - 1}{2 \, \cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1}\right )}{4 \, b} \]
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\[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=- \int \frac {\csc ^{4}{\left (a + b x \right )}}{\csc ^{4}{\left (a + b x \right )} + \sec ^{4}{\left (a + b x \right )}}\, dx - \int \left (- \frac {\sec ^{4}{\left (a + b x \right )}}{\csc ^{4}{\left (a + b x \right )} + \sec ^{4}{\left (a + b x \right )}}\right )\, dx \]
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=-\frac {\sqrt {2} \log \left (\tan \left (b x + a\right )^{2} + \sqrt {2} \tan \left (b x + a\right ) + 1\right ) - \sqrt {2} \log \left (\tan \left (b x + a\right )^{2} - \sqrt {2} \tan \left (b x + a\right ) + 1\right )}{4 \, b} \]
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Time = 0.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.67 \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=\frac {\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sin \left (2 \, b x + 2 \, a\right ) \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \sin \left (2 \, b x + 2 \, a\right ) \right |}}\right )}{4 \, b} \]
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Time = 26.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.32 \[ \int \frac {-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx=-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sin \left (2\,a+2\,b\,x\right )}{2}\right )}{2\,b} \]
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