Optimal. Leaf size=47 \[ \frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.0421372, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3787, 3767, 8, 3768, 3770} \[ \frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3787
Rule 3767
Rule 8
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x)) \, dx &=a \int \sec ^2(c+d x) \, dx+a \int \sec ^3(c+d x) \, dx\\ &=\frac{a \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} a \int \sec (c+d x) \, dx-\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.018978, size = 47, normalized size = 1. \[ \frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 51, normalized size = 1.1 \begin{align*}{\frac{a\tan \left ( dx+c \right ) }{d}}+{\frac{a\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07314, size = 78, normalized size = 1.66 \begin{align*} -\frac{a{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, a \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69099, size = 198, normalized size = 4.21 \begin{align*} \frac{a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{4 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\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 = 1.34646, size = 108, normalized size = 2.3 \begin{align*} \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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