Optimal. Leaf size=35 \[ \frac{(c+d x) \sec ^2(a+b x)}{2 b}-\frac{d \tan (a+b x)}{2 b^2} \]
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Rubi [A] time = 0.0316128, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4409, 3767, 8} \[ \frac{(c+d x) \sec ^2(a+b x)}{2 b}-\frac{d \tan (a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 4409
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (c+d x) \sec ^2(a+b x) \tan (a+b x) \, dx &=\frac{(c+d x) \sec ^2(a+b x)}{2 b}-\frac{d \int \sec ^2(a+b x) \, dx}{2 b}\\ &=\frac{(c+d x) \sec ^2(a+b x)}{2 b}+\frac{d \operatorname{Subst}(\int 1 \, dx,x,-\tan (a+b x))}{2 b^2}\\ &=\frac{(c+d x) \sec ^2(a+b x)}{2 b}-\frac{d \tan (a+b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0621442, size = 48, normalized size = 1.37 \[ -\frac{d \tan (a+b x)}{2 b^2}+\frac{c \sec ^2(a+b x)}{2 b}+\frac{d x \sec ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 61, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ({\frac{bx+a}{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}-{\frac{\tan \left ( bx+a \right ) }{2}} \right ) }-{\frac{ad}{2\,b \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{c}{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.976842, size = 382, normalized size = 10.91 \begin{align*} \frac{c \tan \left (b x + a\right )^{2} - \frac{a d \tan \left (b x + a\right )^{2}}{b} + \frac{2 \,{\left (4 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + 4 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} +{\left (2 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) + 2 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) +{\left (2 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (2 \,{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} b}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.464679, size = 95, normalized size = 2.71 \begin{align*} \frac{b d x - d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + b c}{2 \, b^{2} \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 \left (c + d x\right ) \tan{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23716, size = 771, normalized size = 22.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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