Optimal. Leaf size=114 \[ \frac{(a B+A b) \tan ^3(c+d x)}{3 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a A+3 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.145108, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3997, 3787, 3768, 3770, 3767} \[ \frac{(a B+A b) \tan ^3(c+d x)}{3 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a A+3 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3997
Rule 3787
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int \sec ^3(c+d x) (4 a A+3 b B+4 (A b+a B) \sec (c+d x)) \, dx\\ &=\frac{b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+(A b+a B) \int \sec ^4(c+d x) \, dx+\frac{1}{4} (4 a A+3 b B) \int \sec ^3(c+d x) \, dx\\ &=\frac{(4 a A+3 b B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} (4 a A+3 b B) \int \sec (c+d x) \, dx-\frac{(A b+a B) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(A b+a B) \tan (c+d x)}{d}+\frac{(4 a A+3 b B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{(A b+a B) \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.617527, size = 85, normalized size = 0.75 \[ \frac{3 (4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (8 (a B+A b) (\cos (2 (c+d x))+2) \sec (c+d x)+12 a A+6 b B \sec ^2(c+d x)+9 b B\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 171, normalized size = 1.5 \begin{align*}{\frac{Aa\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{Aa\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,Ba\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Ba \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,Ab\tan \left ( dx+c \right ) }{3\,d}}+{\frac{Ab\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{Bb \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,Bb\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,Bb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981238, size = 220, normalized size = 1.93 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b - 3 \, B b{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A 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 )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.825603, size = 352, normalized size = 3.09 \begin{align*} \frac{3 \,{\left (4 \, A a + 3 \, B b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (4 \, A a + 3 \, B b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (B a + A b\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (4 \, A a + 3 \, B b\right )} \cos \left (d x + c\right )^{2} + 6 \, B b + 8 \,{\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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 ^{3}{\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 = 1.26879, size = 410, normalized size = 3.6 \begin{align*} \frac{3 \,{\left (4 \, A a + 3 \, B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (4 \, A a + 3 \, B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (12 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 12 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, 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} - 40 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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