Optimal. Leaf size=54 \[ \frac{2 \log \left (\tan ^2(a+b x)-\tan (a+b x)+1\right )}{3 b}-\frac{\log (\tan (a+b x)+1)}{3 b}+\frac{\log (\cos (a+b x))}{b} \]
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Rubi [A] time = 0.531603, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {6725, 260, 628} \[ \frac{2 \log \left (\tan ^2(a+b x)-\tan (a+b x)+1\right )}{3 b}-\frac{\log (\tan (a+b x)+1)}{3 b}+\frac{\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 6725
Rule 260
Rule 628
Rubi steps
\begin{align*} \int \frac{-\csc ^3(a+b x)+\sec ^3(a+b x)}{\csc ^3(a+b x)+\sec ^3(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^3}{\left (1+x^2\right ) \left (1+x^3\right )} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{3 (1+x)}-\frac{x}{1+x^2}+\frac{2 (-1+2 x)}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{\log (1+\tan (a+b x))}{3 b}+\frac{2 \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\tan (a+b x)\right )}{3 b}-\frac{\operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\log (\cos (a+b x))}{b}-\frac{\log (1+\tan (a+b x))}{3 b}+\frac{2 \log \left (1-\tan (a+b x)+\tan ^2(a+b x)\right )}{3 b}\\ \end{align*}
Mathematica [A] time = 0.236641, size = 42, normalized size = 0.78 \[ \frac{2 \log (2-\sin (2 (a+b x)))}{3 b}-\frac{\log (\sin (a+b x)+\cos (a+b x))}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.26, size = 56, normalized size = 1. \begin{align*} -{\frac{\ln \left ( 1+\tan \left ( bx+a \right ) \right ) }{3\,b}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( bx+a \right ) \right ) ^{2} \right ) }{2\,b}}+{\frac{2\,\ln \left ( 1-\tan \left ( bx+a \right ) + \left ( \tan \left ( bx+a \right ) \right ) ^{2} \right ) }{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46416, size = 208, normalized size = 3.85 \begin{align*} \frac{2 \, \log \left (-\frac{2 \, \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} + \frac{2 \, \sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{2 \, \sin \left (b x + a\right )^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{\sin \left (b x + a\right )^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1\right ) - \log \left (-\frac{2 \, \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} + \frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right ) - 3 \, \log \left (\frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23174, size = 117, normalized size = 2.17 \begin{align*} -\frac{\log \left (2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right ) - 4 \, \log \left (-\cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2281, size = 70, normalized size = 1.3 \begin{align*} \frac{4 \, \log \left (\tan \left (b x + a\right )^{2} - \tan \left (b x + a\right ) + 1\right ) - 3 \, \log \left (\tan \left (b x + a\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (b x + a\right ) + 1 \right |}\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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