Optimal. Leaf size=35 \[ \frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{b B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.0346277, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3914, 3767, 8, 3770} \[ \frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{b B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3914
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=a A x+(b B) \int \sec ^2(c+d x) \, dx+(A b+a B) \int \sec (c+d x) \, dx\\ &=a A x+\frac{(A b+a B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(b B) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a A x+\frac{(A b+a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0098725, size = 43, normalized size = 1.23 \[ a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 65, normalized size = 1.9 \begin{align*} aAx+{\frac{Ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Aac}{d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Bb\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98273, size = 76, normalized size = 2.17 \begin{align*} \frac{{\left (d x + c\right )} A a + B a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + A b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + B b \tan \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 = 0.503204, size = 225, normalized size = 6.43 \begin{align*} \frac{2 \, A a d x \cos \left (d x + c\right ) +{\left (B a + A b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (B a + A b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B b \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.1132, size = 71, normalized size = 2.03 \begin{align*} \begin{cases} \frac{A a \left (c + d x\right ) + A b \log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )} + B a \log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )} + B b \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \sec{\left (c \right )}\right ) \left (a + b \sec{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21419, size = 113, normalized size = 3.23 \begin{align*} \frac{{\left (d x + c\right )} A a +{\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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