Optimal. Leaf size=60 \[ \frac{a^2 B \sin (c+d x)}{d}+\frac{b (2 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a C+2 b B)+\frac{b^2 C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.176471, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4072, 4024, 3770, 3767, 8} \[ \frac{a^2 B \sin (c+d x)}{d}+\frac{b (2 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a C+2 b B)+\frac{b^2 C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4024
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos (c+d x) (a+b \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx\\ &=\frac{a^2 B \sin (c+d x)}{d}-\int \left (-a (2 b B+a C)+\left (-b^2 B-2 a b C\right ) \sec (c+d x)-b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=a (2 b B+a C) x+\frac{a^2 B \sin (c+d x)}{d}+\left (b^2 C\right ) \int \sec ^2(c+d x) \, dx+(b (b B+2 a C)) \int \sec (c+d x) \, dx\\ &=a (2 b B+a C) x+\frac{b (b B+2 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 B \sin (c+d x)}{d}-\frac{\left (b^2 C\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a (2 b B+a C) x+\frac{b (b B+2 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 B \sin (c+d x)}{d}+\frac{b^2 C \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.49343, size = 109, normalized size = 1.82 \[ \frac{a^2 B \sin (c+d x)+a (c+d x) (a C+2 b B)-b (2 a C+b B) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+b (2 a C+b B) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+b^2 C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 104, normalized size = 1.7 \begin{align*} 2\,Babx+{a}^{2}Cx+{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{B{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{Babc}{d}}+{\frac{{b}^{2}C\tan \left ( dx+c \right ) }{d}}+2\,{\frac{abC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{C{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972229, size = 139, normalized size = 2.32 \begin{align*} \frac{2 \,{\left (d x + c\right )} C a^{2} + 4 \,{\left (d x + c\right )} B a b + 2 \, C a b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a^{2} \sin \left (d x + c\right ) + 2 \, C b^{2} \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.522169, size = 294, normalized size = 4.9 \begin{align*} \frac{2 \,{\left (C a^{2} + 2 \, B a b\right )} d x \cos \left (d x + c\right ) +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (B a^{2} \cos \left (d x + c\right ) + C b^{2}\right )} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22587, size = 208, normalized size = 3.47 \begin{align*} \frac{{\left (C a^{2} + 2 \, B a b\right )}{\left (d x + c\right )} +{\left (2 \, C a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, C a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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