Optimal. Leaf size=15 \[ \frac{\sec ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0194794, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2606, 30} \[ \frac{\sec ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \sec ^2(a+b x) \tan (a+b x) \, dx &=\frac{\operatorname{Subst}(\int x \, dx,x,\sec (a+b x))}{b}\\ &=\frac{\sec ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.010365, size = 15, normalized size = 1. \[ \frac{\sec ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sec \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989576, size = 23, normalized size = 1.53 \begin{align*} -\frac{1}{2 \,{\left (\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 = 1.67507, size = 32, normalized size = 2.13 \begin{align*} \frac{1}{2 \, b \cos \left (b x + a\right )^{2}} \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 \sin{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15784, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2 \, b \cos \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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