Optimal. Leaf size=95 \[ \frac{a (A+C) \sin (c+d x)}{d}+\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a A \sin ^3(c+d x)}{3 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C) \]
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Rubi [A] time = 0.177301, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4075, 4047, 2635, 8, 4044, 3013} \[ \frac{a (A+C) \sin (c+d x)}{d}+\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a A \sin ^3(c+d x)}{3 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C) \]
Antiderivative was successfully verified.
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Rule 4075
Rule 4047
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+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 A-a (3 A+4 C) \sec (c+d x)-4 a 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 A-4 a C \sec ^2(c+d x)\right ) \, dx+\frac{1}{4} (a (3 A+4 C)) \int \cos ^2(c+d x) \, dx\\ &=\frac{a (3 A+4 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 a C-4 a A \cos ^2(c+d x)\right ) \, dx+\frac{1}{8} (a (3 A+4 C)) \int 1 \, dx\\ &=\frac{1}{8} a (3 A+4 C) x+\frac{a (3 A+4 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 A-4 a C+4 a A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d}\\ &=\frac{1}{8} a (3 A+4 C) x+\frac{a (A+C) \sin (c+d x)}{d}+\frac{a (3 A+4 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 A \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.230785, size = 77, normalized size = 0.81 \[ \frac{a (24 (3 A+4 C) \sin (c+d x)+24 (A+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+36 A c+36 A d x+48 c C+48 C d x)}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 96, normalized size = 1. \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{Aa \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aC\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929224, size = 122, normalized size = 1.28 \begin{align*} -\frac{32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 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 - 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 96 \, C a \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 = 0.496195, size = 189, normalized size = 1.99 \begin{align*} \frac{3 \,{\left (3 \, A + 4 \, C\right )} a d x +{\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, A a \cos \left (d x + c\right )^{2} + 3 \,{\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right ) + 8 \,{\left (2 \, A + 3 \, C\right )} a\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 [A] time = 1.15484, size = 211, normalized size = 2.22 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, C a\right )}{\left (d x + c\right )} + \frac{2 \,{\left (9 \, A 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} + 49 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 60 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 31 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 84 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 39 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 36 \, C a \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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