Optimal. Leaf size=31 \[ \frac{\sec ^8(a+b x)}{8 b}-\frac{\sec ^6(a+b x)}{6 b} \]
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Rubi [A] time = 0.0321571, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2606, 14} \[ \frac{\sec ^8(a+b x)}{8 b}-\frac{\sec ^6(a+b x)}{6 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 14
Rubi steps
\begin{align*} \int \sec ^6(a+b x) \tan ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^5 \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-x^5+x^7\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac{\sec ^6(a+b x)}{6 b}+\frac{\sec ^8(a+b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.0415676, size = 28, normalized size = 0.9 \[ -\frac{4 \sec ^6(a+b x)-3 \sec ^8(a+b x)}{24 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 60, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{12\, \left ( \cos \left ( bx+a \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{24\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03137, size = 80, normalized size = 2.58 \begin{align*} \frac{4 \, \sin \left (b x + a\right )^{2} - 1}{24 \,{\left (\sin \left (b x + a\right )^{8} - 4 \, \sin \left (b x + a\right )^{6} + 6 \, \sin \left (b x + a\right )^{4} - 4 \, \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 = 1.6317, size = 66, normalized size = 2.13 \begin{align*} -\frac{4 \, \cos \left (b x + a\right )^{2} - 3}{24 \, b \cos \left (b x + a\right )^{8}} \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 = 1.2006, size = 34, normalized size = 1.1 \begin{align*} -\frac{4 \, \cos \left (b x + a\right )^{2} - 3}{24 \, b \cos \left (b x + a\right )^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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