Optimal. Leaf size=116 \[ \frac{\sin (c+d x) (a B+A b+b C)}{d}+\frac{\sin (c+d x) \cos (c+d x) (3 a A+4 a C+4 b B)}{8 d}-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{1}{8} x (3 a A+4 a C+4 b B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.214684, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4074, 4047, 2635, 8, 4044, 3013} \[ \frac{\sin (c+d x) (a B+A b+b C)}{d}+\frac{\sin (c+d x) \cos (c+d x) (3 a A+4 a C+4 b B)}{8 d}-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{1}{8} x (3 a A+4 a C+4 b B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 4074
Rule 4047
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{1}{4} \int \cos ^3(c+d x) \left (-4 (A b+a B)-(3 a A+4 b B+4 a C) \sec (c+d x)-4 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{1}{4} \int \cos ^3(c+d x) \left (-4 (A b+a B)-4 b C \sec ^2(c+d x)\right ) \, dx-\frac{1}{4} (-3 a A-4 b B-4 a C) \int \cos ^2(c+d x) \, dx\\ &=\frac{(3 a A+4 b B+4 a C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{1}{4} \int \cos (c+d x) \left (-4 b C-4 (A b+a B) \cos ^2(c+d x)\right ) \, dx-\frac{1}{8} (-3 a A-4 b B-4 a C) \int 1 \, dx\\ &=\frac{1}{8} (3 a A+4 b B+4 a C) x+\frac{(3 a A+4 b B+4 a C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a A \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{\operatorname{Subst}\left (\int \left (-4 (A b+a B)-4 b C+4 (A b+a B) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d}\\ &=\frac{1}{8} (3 a A+4 b B+4 a C) x+\frac{(A b+a B+b C) \sin (c+d x)}{d}+\frac{(3 a A+4 b B+4 a C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{(A b+a B) \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.315989, size = 117, normalized size = 1.01 \[ \frac{24 \sin (c+d x) (3 a B+3 A b+4 b C)+24 \sin (2 (c+d x)) (a (A+C)+b B)+3 a A \sin (4 (c+d x))+36 a A c+36 a A d x+8 a B \sin (3 (c+d x))+48 a c C+48 a C d x+8 A b \sin (3 (c+d x))+48 b B c+48 b B d x}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 141, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( Aa \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{Ab \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{Ba \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Bb \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +C\sin \left ( dx+c \right ) b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03443, size = 178, normalized size = 1.53 \begin{align*} \frac{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 )} A a - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b + 96 \, C b \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 = 0.518022, size = 239, normalized size = 2.06 \begin{align*} \frac{3 \,{\left ({\left (3 \, A + 4 \, C\right )} a + 4 \, B b\right )} d x +{\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \,{\left (B a + A b\right )} \cos \left (d x + c\right )^{2} + 16 \, B a + 8 \,{\left (2 \, A + 3 \, C\right )} b + 3 \,{\left ({\left (3 \, A + 4 \, C\right )} a + 4 \, B b\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 = 1.16558, size = 529, normalized size = 4.56 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, C a + 4 \, B b\right )}{\left (d x + c\right )} - \frac{2 \,{\left (15 \, 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} + 12 \, C 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} + 12 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 9 \, 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} + 12 \, C 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} + 12 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 72 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, 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} - 12 \, C 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} - 12 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 72 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, 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 ) - 12 \, C 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 ) - 12 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 24 \, C 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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