Optimal. Leaf size=31 \[ \frac{\sec ^7(a+b x)}{7 b}-\frac{\sec ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0321572, 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} \[ \frac{\sec ^7(a+b x)}{7 b}-\frac{\sec ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 14
Rubi steps
\begin{align*} \int \sec ^5(a+b x) \tan ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac{\sec ^5(a+b x)}{5 b}+\frac{\sec ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0282945, size = 31, normalized size = 1. \[ \frac{\sec ^7(a+b x)}{7 b}-\frac{\sec ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 96, normalized size = 3.1 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{7\, \left ( \cos \left ( bx+a \right ) \right ) ^{7}}}+{\frac{3\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{35\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{35\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{35\,\cos \left ( bx+a \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{35}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01664, size = 34, normalized size = 1.1 \begin{align*} -\frac{7 \, \cos \left (b x + a\right )^{2} - 5}{35 \, b \cos \left (b x + a\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67524, size = 66, normalized size = 2.13 \begin{align*} -\frac{7 \, \cos \left (b x + a\right )^{2} - 5}{35 \, b \cos \left (b x + a\right )^{7}} \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 = 1.17587, size = 34, normalized size = 1.1 \begin{align*} -\frac{7 \, \cos \left (b x + a\right )^{2} - 5}{35 \, b \cos \left (b x + a\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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