Optimal. Leaf size=46 \[ \frac{\sec ^{12}(a+b x)}{12 b}-\frac{\sec ^{10}(a+b x)}{5 b}+\frac{\sec ^8(a+b x)}{8 b} \]
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Rubi [A] time = 0.0402263, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2606, 266, 43} \[ \frac{\sec ^{12}(a+b x)}{12 b}-\frac{\sec ^{10}(a+b x)}{5 b}+\frac{\sec ^8(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \sec ^8(a+b x) \tan ^5(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^7 \left (-1+x^2\right )^2 \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int (-1+x)^2 x^3 \, dx,x,\sec ^2(a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\sec ^2(a+b x)\right )}{2 b}\\ &=\frac{\sec ^8(a+b x)}{8 b}-\frac{\sec ^{10}(a+b x)}{5 b}+\frac{\sec ^{12}(a+b x)}{12 b}\\ \end{align*}
Mathematica [A] time = 0.128708, size = 38, normalized size = 0.83 \[ \frac{10 \sec ^{12}(a+b x)-24 \sec ^{10}(a+b x)+15 \sec ^8(a+b x)}{120 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 78, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{12\, \left ( \cos \left ( bx+a \right ) \right ) ^{12}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{20\, \left ( \cos \left ( bx+a \right ) \right ) ^{10}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{40\, \left ( \cos \left ( bx+a \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{6}}{120\, \left ( \cos \left ( bx+a \right ) \right ) ^{6}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.974354, size = 120, normalized size = 2.61 \begin{align*} \frac{15 \, \sin \left (b x + a\right )^{4} - 6 \, \sin \left (b x + a\right )^{2} + 1}{120 \,{\left (\sin \left (b x + a\right )^{12} - 6 \, \sin \left (b x + a\right )^{10} + 15 \, \sin \left (b x + a\right )^{8} - 20 \, \sin \left (b x + a\right )^{6} + 15 \, \sin \left (b x + a\right )^{4} - 6 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64175, size = 97, normalized size = 2.11 \begin{align*} \frac{15 \, \cos \left (b x + a\right )^{4} - 24 \, \cos \left (b x + a\right )^{2} + 10}{120 \, b \cos \left (b x + a\right )^{12}} \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 [B] time = 1.20258, size = 247, normalized size = 5.37 \begin{align*} -\frac{32 \,{\left (\frac{5 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac{15 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac{39 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac{42 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac{39 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac{15 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac{5 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}}\right )}}{15 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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