Optimal. Leaf size=207 \[ \frac{\sin (c+d x) \left (16 a^2 b B+3 a^3 C+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac{b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac{1}{8} x \left (12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right )+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.563557, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {3049, 3033, 3023, 2735, 3770} \[ \frac{\sin (c+d x) \left (16 a^2 b B+3 a^3 C+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac{b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac{1}{8} x \left (12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right )+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 3049
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{4} \int (a+b \cos (c+d x))^2 \left (4 a A+(4 A b+4 a B+3 b C) \cos (c+d x)+(4 b B+3 a C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{12} \int (a+b \cos (c+d x)) \left (12 a^2 A+\left (24 a A b+12 a^2 B+8 b^2 B+15 a b C\right ) \cos (c+d x)+\left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{24} \int \left (24 a^3 A+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \cos (c+d x)+4 \left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac{b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{24} \int \left (24 a^3 A+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) x+\frac{\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac{b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) x+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac{b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.934392, size = 218, normalized size = 1.05 \[ \frac{12 (c+d x) \left (12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right )+24 \sin (c+d x) \left (12 a^2 b B+4 a^3 C+3 a b^2 (4 A+3 C)+3 b^3 B\right )+24 b \sin (2 (c+d x)) \left (3 a^2 C+3 a b B+A b^2+b^2 C\right )-96 a^3 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+96 a^3 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 b^2 (3 a C+b B) \sin (3 (c+d x))+3 b^3 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 362, normalized size = 1.8 \begin{align*}{\frac{{a}^{3}C\sin \left ( dx+c \right ) }{d}}+3\,A{a}^{2}bx+{\frac{A{b}^{3}c}{2\,d}}+{\frac{3\,C{b}^{3}c}{8\,d}}+{\frac{3\,{a}^{2}bCx}{2}}+{\frac{2\,{b}^{3}B\sin \left ( dx+c \right ) }{3\,d}}+{\frac{3\,a{b}^{2}Bx}{2}}+{\frac{B{a}^{3}c}{d}}+{\frac{A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{b}^{3}Cx}{8}}+{\frac{3\,a{b}^{2}Bc}{2\,d}}+{\frac{3\,C{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+2\,{\frac{Ca{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{3\,{a}^{2}bCc}{2\,d}}+{\frac{C{b}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+3\,{\frac{aA{b}^{2}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{2}bB\sin \left ( dx+c \right ) }{d}}+{\frac{B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{3\,d}}+{\frac{A\cos \left ( dx+c \right ){b}^{3}\sin \left ( dx+c \right ) }{2\,d}}+3\,{\frac{A{a}^{2}bc}{d}}+{a}^{3}Bx+{\frac{3\,B\cos \left ( dx+c \right ) a{b}^{2}\sin \left ( dx+c \right ) }{2\,d}}+{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}a{b}^{2}}{d}}+{\frac{3\,C\cos \left ( dx+c \right ){a}^{2}b\sin \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{3}x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994961, size = 323, normalized size = 1.56 \begin{align*} \frac{96 \,{\left (d x + c\right )} B a^{3} + 288 \,{\left (d x + c\right )} A a^{2} b + 72 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 72 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} - 96 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{2} + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{3} + 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 96 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, C a^{3} \sin \left (d x + c\right ) + 288 \, B a^{2} b \sin \left (d x + c\right ) + 288 \, A a b^{2} \sin \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9642, size = 463, normalized size = 2.24 \begin{align*} \frac{12 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (8 \, B a^{3} + 12 \,{\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} +{\left (4 \, A + 3 \, C\right )} b^{3}\right )} d x +{\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 24 \, C a^{3} + 72 \, B a^{2} b + 24 \,{\left (3 \, A + 2 \, C\right )} a b^{2} + 16 \, B b^{3} + 8 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (12 \, C a^{2} b + 12 \, B a b^{2} +{\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \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 = 1.22459, size = 976, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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