Optimal. Leaf size=93 \[ \frac{(3 a A+2 b B) \tan (c+d x)}{3 d}+\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b B \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.132709, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {3997, 3787, 3767, 8, 3768, 3770} \[ \frac{(3 a A+2 b B) \tan (c+d x)}{3 d}+\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b B \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3997
Rule 3787
Rule 3767
Rule 8
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{b B \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int \sec ^2(c+d x) (3 a A+2 b B+3 (A b+a B) \sec (c+d x)) \, dx\\ &=\frac{b B \sec ^2(c+d x) \tan (c+d x)}{3 d}+(A b+a B) \int \sec ^3(c+d x) \, dx+\frac{1}{3} (3 a A+2 b B) \int \sec ^2(c+d x) \, dx\\ &=\frac{(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac{b B \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} (A b+a B) \int \sec (c+d x) \, dx-\frac{(3 a A+2 b B) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{(A b+a B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(3 a A+2 b B) \tan (c+d x)}{3 d}+\frac{(A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac{b B \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.268569, size = 67, normalized size = 0.72 \[ \frac{3 (a B+A b) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (3 (a B+A b) \sec (c+d x)+6 a A+2 b B \tan ^2(c+d x)+6 b B\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 128, normalized size = 1.4 \begin{align*}{\frac{Aa\tan \left ( dx+c \right ) }{d}}+{\frac{B\sec \left ( dx+c \right ) a\tan \left ( dx+c \right ) }{2\,d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{A\sec \left ( dx+c \right ) b\tan \left ( dx+c \right ) }{2\,d}}+{\frac{Ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,Bb\tan \left ( dx+c \right ) }{3\,d}}+{\frac{B \left ( \sec \left ( dx+c \right ) \right ) ^{2}b\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 = 0.974597, size = 171, normalized size = 1.84 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 3 \, 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 )} - 3 \, A 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 \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 = 0.979597, size = 298, normalized size = 3.2 \begin{align*} \frac{3 \,{\left (B a + A b\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (B a + A b\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \,{\left (3 \, A a + 2 \, B b\right )} \cos \left (d x + c\right )^{2} + 2 \, B b + 3 \,{\left (B a + A 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 )}\right ) \sec ^{2}{\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 = 1.24908, size = 284, normalized size = 3.05 \begin{align*} \frac{3 \,{\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\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 \,{\left (6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, 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 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, B 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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