Optimal. Leaf size=72 \[ \frac{\log \left (\tan ^2(a+b x)+\sqrt{2} \tan (a+b x)+1\right )}{2 \sqrt{2} b}-\frac{\log \left (\tan ^2(a+b x)-\sqrt{2} \tan (a+b x)+1\right )}{2 \sqrt{2} b} \]
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Rubi [A] time = 0.151379, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {1165, 628} \[ \frac{\log \left (\tan ^2(a+b x)+\sqrt{2} \tan (a+b x)+1\right )}{2 \sqrt{2} b}-\frac{\log \left (\tan ^2(a+b x)-\sqrt{2} \tan (a+b x)+1\right )}{2 \sqrt{2} b} \]
Antiderivative was successfully verified.
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Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\cos ^4(a+b x)-\sin ^4(a+b x)}{\cos ^4(a+b x)+\sin ^4(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{\operatorname{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{\operatorname{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*}
Mathematica [A] time = 0.0345226, size = 25, normalized size = 0.35 \[ \frac{\tanh ^{-1}\left (\frac{\sin (2 a+2 b x)}{\sqrt{2}}\right )}{\sqrt{2} b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 108, normalized size = 1.5 \begin{align*}{\frac{\sqrt{2}}{8\,b}\ln \left ({\frac{1+\sqrt{2}\tan \left ( bx+a \right ) + \left ( \tan \left ( bx+a \right ) \right ) ^{2}}{1-\sqrt{2}\tan \left ( bx+a \right ) + \left ( \tan \left ( bx+a \right ) \right ) ^{2}}} \right ) }-{\frac{\sqrt{2}}{8\,b}\ln \left ({\frac{1-\sqrt{2}\tan \left ( bx+a \right ) + \left ( \tan \left ( bx+a \right ) \right ) ^{2}}{1+\sqrt{2}\tan \left ( bx+a \right ) + \left ( \tan \left ( bx+a \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43376, size = 78, normalized size = 1.08 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12544, size = 193, normalized size = 2.68 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 28.038, size = 122, normalized size = 1.69 \begin{align*} \begin{cases} - \frac{\sqrt{2} \log{\left (4 \sin ^{2}{\left (a + b x \right )} - 4 \sqrt{2} \sin{\left (a + b x \right )} \cos{\left (a + b x \right )} + 4 \cos ^{2}{\left (a + b x \right )} \right )}}{4 b} + \frac{\sqrt{2} \log{\left (4 \sin ^{2}{\left (a + b x \right )} + 4 \sqrt{2} \sin{\left (a + b x \right )} \cos{\left (a + b x \right )} + 4 \cos ^{2}{\left (a + b x \right )} \right )}}{4 b} & \text{for}\: b \neq 0 \\\frac{x \left (- \sin ^{4}{\left (a \right )} + \cos ^{4}{\left (a \right )}\right )}{\sin ^{4}{\left (a \right )} + \cos ^{4}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2328, size = 65, normalized size = 0.9 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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