Optimal. Leaf size=101 \[ \frac{\tan (c+d x) (3 a B+3 A b+2 b C)}{3 d}+\frac{(a (2 A+C)+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.13822, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4076, 4047, 3767, 8, 4046, 3770} \[ \frac{\tan (c+d x) (3 a B+3 A b+2 b C)}{3 d}+\frac{(a (2 A+C)+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4076
Rule 4047
Rule 3767
Rule 8
Rule 4046
Rule 3770
Rubi steps
\begin{align*} \int \sec (c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{b C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int \sec (c+d x) \left (3 a A+(3 A b+3 a B+2 b C) \sec (c+d x)+3 (b B+a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{b C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int \sec (c+d x) \left (3 a A+3 (b B+a C) \sec ^2(c+d x)\right ) \, dx+\frac{1}{3} (3 A b+3 a B+2 b C) \int \sec ^2(c+d x) \, dx\\ &=\frac{(b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac{b C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} (b B+a (2 A+C)) \int \sec (c+d x) \, dx-\frac{(3 A b+3 a B+2 b C) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{(b B+a (2 A+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(3 A b+3 a B+2 b C) \tan (c+d x)}{3 d}+\frac{(b B+a C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac{b C \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.370176, size = 75, normalized size = 0.74 \[ \frac{3 (a (2 A+C)+b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (3 (a C+b B) \sec (c+d x)+6 a B+6 A b+2 b C \tan ^2(c+d x)+6 b C\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 160, normalized size = 1.6 \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{aC\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{Ab\tan \left ( dx+c \right ) }{d}}+{\frac{Bb\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{2\,d}}+{\frac{Bb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,Cb\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Cb \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05017, size = 209, normalized size = 2.07 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b - 3 \, C 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 )} - 3 \, B b{\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 )} + 12 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, B a \tan \left (d x + c\right ) + 12 \, A b \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.597705, size = 331, normalized size = 3.28 \begin{align*} \frac{3 \,{\left ({\left (2 \, A + C\right )} a + B b\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left ({\left (2 \, A + C\right )} a + B b\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \,{\left (3 \, B a +{\left (3 \, A + 2 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 2 \, C b + 3 \,{\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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 (a + b \sec{\left (c + d x \right )}\right ) \left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25643, size = 352, normalized size = 3.49 \begin{align*} \frac{3 \,{\left (2 \, A a + C a + B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (2 \, A a + C a + B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C b \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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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