Optimal. Leaf size=27 \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x \]
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Rubi [A] time = 0.0465878, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2838, 2606, 8, 3473} \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2606
Rule 8
Rule 3473
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec (c+d x) \tan (c+d x) \, dx+a \int \tan ^2(c+d x) \, dx\\ &=\frac{a \tan (c+d x)}{d}-a \int 1 \, dx+\frac{a \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-a x+\frac{a \sec (c+d x)}{d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0197155, size = 36, normalized size = 1.33 \[ -\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 32, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( a \left ( \tan \left ( dx+c \right ) -dx-c \right ) +{\frac{a}{\cos \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65113, size = 43, normalized size = 1.59 \begin{align*} -\frac{{\left (d x + c - \tan \left (d x + c\right )\right )} a - \frac{a}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.06085, size = 143, normalized size = 5.3 \begin{align*} -\frac{a d x +{\left (a d x - a\right )} \cos \left (d x + c\right ) -{\left (a d x + a\right )} \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \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*} a \left (\int \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28639, size = 39, normalized size = 1.44 \begin{align*} -\frac{{\left (d x + c\right )} a + \frac{2 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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