Optimal. Leaf size=41 \[ \frac{\sec ^5(a+b x)}{5 b}-\frac{2 \sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b} \]
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Rubi [A] time = 0.023039, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2606, 194} \[ \frac{\sec ^5(a+b x)}{5 b}-\frac{2 \sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \sec (a+b x) \tan ^5(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac{\sec (a+b x)}{b}-\frac{2 \sec ^3(a+b x)}{3 b}+\frac{\sec ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.027876, size = 41, normalized size = 1. \[ \frac{\sec ^5(a+b x)}{5 b}-\frac{2 \sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 88, normalized size = 2.2 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{5\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}}-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{15\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{5\,\cos \left ( bx+a \right ) }}+{\frac{\cos \left ( bx+a \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00889, size = 47, normalized size = 1.15 \begin{align*} \frac{15 \, \cos \left (b x + a\right )^{4} - 10 \, \cos \left (b x + a\right )^{2} + 3}{15 \, b \cos \left (b x + a\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58203, size = 93, normalized size = 2.27 \begin{align*} \frac{15 \, \cos \left (b x + a\right )^{4} - 10 \, \cos \left (b x + a\right )^{2} + 3}{15 \, b \cos \left (b x + a\right )^{5}} \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 = 1.15996, size = 97, normalized size = 2.37 \begin{align*} \frac{16 \,{\left (\frac{5 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{10 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}}{15 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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