Optimal. Leaf size=39 \[ \frac{\tan ^4(a+b x)}{4 b}+\frac{\tan ^2(a+b x)}{b}+\frac{\log (\tan (a+b x))}{b} \]
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Rubi [A] time = 0.0262298, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2620, 266, 43} \[ \frac{\tan ^4(a+b x)}{4 b}+\frac{\tan ^2(a+b x)}{b}+\frac{\log (\tan (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \csc (a+b x) \sec ^5(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^2}{x} \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (2+\frac{1}{x}+x\right ) \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=\frac{\log (\tan (a+b x))}{b}+\frac{\tan ^2(a+b x)}{b}+\frac{\tan ^4(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0907424, size = 46, normalized size = 1.18 \[ -\frac{-\sec ^4(a+b x)-2 \sec ^2(a+b x)-4 \log (\sin (a+b x))+4 \log (\cos (a+b x))}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 39, normalized size = 1. \begin{align*}{\frac{1}{4\,b \left ( \cos \left ( bx+a \right ) \right ) ^{4}}}+{\frac{1}{2\,b \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974171, size = 88, normalized size = 2.26 \begin{align*} -\frac{\frac{2 \, \sin \left (b x + a\right )^{2} - 3}{\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2} + 1} + 2 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 2 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74276, size = 185, normalized size = 4.74 \begin{align*} -\frac{2 \, \cos \left (b x + a\right )^{4} \log \left (\cos \left (b x + a\right )^{2}\right ) - 2 \, \cos \left (b x + a\right )^{4} \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\right ) - 2 \, \cos \left (b x + a\right )^{2} - 1}{4 \, b \cos \left (b x + a\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{5}{\left (a + b x \right )}}{\sin{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19017, size = 230, normalized size = 5.9 \begin{align*} \frac{\frac{\frac{52 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{102 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{52 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{25 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 25}{{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{4}} + 6 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 12 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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