Optimal. Leaf size=131 \[ -\frac{a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \sin (c+d x)}{5 d}+\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C) \]
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Rubi [A] time = 0.186607, antiderivative size = 131, 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, 2633, 4045, 2635, 8} \[ -\frac{a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \sin (c+d x)}{5 d}+\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C) \]
Antiderivative was successfully verified.
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Rule 4075
Rule 4047
Rule 2633
Rule 4045
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{1}{5} \int \cos ^4(c+d x) \left (-5 a A-a (4 A+5 C) \sec (c+d x)-5 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{1}{5} \int \cos ^4(c+d x) \left (-5 a A-5 a C \sec ^2(c+d x)\right ) \, dx+\frac{1}{5} (a (4 A+5 C)) \int \cos ^3(c+d x) \, dx\\ &=\frac{a A \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{a A \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac{1}{4} (a (3 A+4 C)) \int \cos ^2(c+d x) \, dx-\frac{(a (4 A+5 C)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{a (4 A+5 C) \sin (c+d x)}{5 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 \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac{1}{8} (a (3 A+4 C)) \int 1 \, dx\\ &=\frac{1}{8} a (3 A+4 C) x+\frac{a (4 A+5 C) \sin (c+d x)}{5 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 \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{a (4 A+5 C) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.278168, size = 86, normalized size = 0.66 \[ \frac{a \left (-160 (2 A+C) \sin ^3(c+d x)+480 (A+C) \sin (c+d x)+15 (4 (3 A+4 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+A \sin (4 (c+d x)))+96 A \sin ^5(c+d x)\right )}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 117, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{Aa\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+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{aC \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.933367, size = 153, normalized size = 1.17 \begin{align*} \frac{32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a + 15 \,{\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 - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.500726, size = 242, normalized size = 1.85 \begin{align*} \frac{15 \,{\left (3 \, A + 4 \, C\right )} a d x +{\left (24 \, A a \cos \left (d x + c\right )^{4} + 30 \, A a \cos \left (d x + c\right )^{3} + 8 \,{\left (4 \, A + 5 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \,{\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right ) + 16 \,{\left (4 \, A + 5 \, C\right )} a\right )} \sin \left (d x + c\right )}{120 \, 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 = 1.1905, size = 251, normalized size = 1.92 \begin{align*} \frac{15 \,{\left (3 \, A a + 4 \, C a\right )}{\left (d x + c\right )} + \frac{2 \,{\left (45 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 60 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 130 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 200 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 464 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 400 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 190 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 440 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 195 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 180 \, 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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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