Optimal. Leaf size=86 \[ \frac{1}{2} x \left (2 a^2 B+4 a A b+b^2 B\right )+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a B+2 A b) \sin (c+d x)}{2 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))}{2 d} \]
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Rubi [A] time = 0.176197, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2990, 3023, 2735, 3770} \[ \frac{1}{2} x \left (2 a^2 B+4 a A b+b^2 B\right )+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a B+2 A b) \sin (c+d x)}{2 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 2990
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx &=\frac{b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\frac{1}{2} \int \left (2 a^2 A+\left (4 a A b+2 a^2 B+b^2 B\right ) \cos (c+d x)+b (2 A b+3 a B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac{b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\frac{1}{2} \int \left (2 a^2 A+\left (4 a A b+2 a^2 B+b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (4 a A b+2 a^2 B+b^2 B\right ) x+\frac{b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac{b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} \left (4 a A b+2 a^2 B+b^2 B\right ) x+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (2 A b+3 a B) \sin (c+d x)}{2 d}+\frac{b B (a+b \cos (c+d x)) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.221827, size = 120, normalized size = 1.4 \[ \frac{2 (c+d x) \left (2 a^2 B+4 a A b+b^2 B\right )-4 a^2 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 a^2 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 b (2 a B+A b) \sin (c+d x)+b^2 B \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 120, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{a}^{2}Bx+{\frac{B{a}^{2}c}{d}}+2\,Aabx+2\,{\frac{Aabc}{d}}+2\,{\frac{Bab\sin \left ( dx+c \right ) }{d}}+{\frac{A{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{{b}^{2}B\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{2}Bx}{2}}+{\frac{{b}^{2}Bc}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04364, size = 124, normalized size = 1.44 \begin{align*} \frac{4 \,{\left (d x + c\right )} B a^{2} + 8 \,{\left (d x + c\right )} A a b +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{2} + 4 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 8 \, B a b \sin \left (d x + c\right ) + 4 \, A b^{2} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54185, size = 213, normalized size = 2.48 \begin{align*} \frac{A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )} d x +{\left (B b^{2} \cos \left (d x + c\right ) + 4 \, B a b + 2 \, A b^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \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 \cos{\left (c + d x \right )}\right ) \left (a + b \cos{\left (c + d x \right )}\right )^{2} \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 = 1.297, size = 240, normalized size = 2.79 \begin{align*} \frac{2 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, A a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) +{\left (2 \, B a^{2} + 4 \, A a b + B b^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (4 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + B b^{2} \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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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