Optimal. Leaf size=56 \[ \frac{a (A+B) \tan (c+d x)}{d}+\frac{a (2 A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.0673097, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3997, 3787, 3770, 3767, 8} \[ \frac{a (A+B) \tan (c+d x)}{d}+\frac{a (2 A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3997
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{a B \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int \sec (c+d x) (a (2 A+B)+2 a (A+B) \sec (c+d x)) \, dx\\ &=\frac{a B \sec (c+d x) \tan (c+d x)}{2 d}+(a (A+B)) \int \sec ^2(c+d x) \, dx+\frac{1}{2} (a (2 A+B)) \int \sec (c+d x) \, dx\\ &=\frac{a (2 A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \sec (c+d x) \tan (c+d x)}{2 d}-\frac{(a (A+B)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{a (2 A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a B \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0258427, size = 75, normalized size = 1.34 \[ \frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 86, normalized size = 1.5 \begin{align*}{\frac{Aa\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Ba\tan \left ( dx+c \right ) }{d}}+{\frac{Aa\tan \left ( dx+c \right ) }{d}}+{\frac{Ba\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Ba\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 = 0.964841, size = 119, normalized size = 2.12 \begin{align*} -\frac{B 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 a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 4 \, A a \tan \left (d x + c\right ) - 4 \, B 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 = 0.47976, size = 239, normalized size = 4.27 \begin{align*} \frac{{\left (2 \, A + B\right )} a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, A + B\right )} a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \,{\left (A + B\right )} a \cos \left (d x + c\right ) + B 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 A \sec{\left (c + d x \right )}\, dx + \int A \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int B \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 [B] time = 1.34501, size = 167, normalized size = 2.98 \begin{align*} \frac{{\left (2 \, A a + B a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, A a + B a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (2 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, B 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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