Optimal. Leaf size=28 \[ \frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0284686, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3486, 3767, 8} \[ \frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx &=\frac{b \sec ^2(c+d x)}{2 d}+a \int \sec ^2(c+d x) \, dx\\ &=\frac{b \sec ^2(c+d x)}{2 d}-\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{b \sec ^2(c+d x)}{2 d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0133805, size = 28, normalized size = 1. \[ \frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 25, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{b}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+a\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5035, size = 27, normalized size = 0.96 \begin{align*} \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}{2 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76126, size = 81, normalized size = 2.89 \begin{align*} \frac{2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{2 \, d \cos \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.19389, size = 34, normalized size = 1.21 \begin{align*} \begin{cases} \frac{a \tan{\left (c + d x \right )} + \frac{b \tan ^{2}{\left (c + d x \right )}}{2}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \sec ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2868, size = 34, normalized size = 1.21 \begin{align*} \frac{b \tan \left (d x + c\right )^{2} + 2 \, a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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