Optimal. Leaf size=279 \[ -\frac{2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac{18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^9(c+d x)}{9 d}+\frac{a^4 \sin ^9(c+d x)}{9 d}-\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{6 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^9(c+d x)}{9 d}-\frac{4 a b^3 \cos ^7(c+d x)}{7 d}+\frac{b^4 \sin ^9(c+d x)}{9 d}-\frac{2 b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.258005, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3090, 2633, 2565, 30, 2564, 270, 14} \[ -\frac{2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac{18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^9(c+d x)}{9 d}+\frac{a^4 \sin ^9(c+d x)}{9 d}-\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{6 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^9(c+d x)}{9 d}-\frac{4 a b^3 \cos ^7(c+d x)}{7 d}+\frac{b^4 \sin ^9(c+d x)}{9 d}-\frac{2 b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 2633
Rule 2565
Rule 30
Rule 2564
Rule 270
Rule 14
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \cos ^9(c+d x)+4 a^3 b \cos ^8(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^7(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^6(c+d x) \sin ^3(c+d x)+b^4 \cos ^5(c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^9(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^8(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^7(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac{a^4 \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int x^8 \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^4 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{4 a^3 b \cos ^9(c+d x)}{9 d}+\frac{a^4 \sin (c+d x)}{d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{6 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{a^4 \sin ^9(c+d x)}{9 d}+\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^4 \operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{4 a b^3 \cos ^7(c+d x)}{7 d}-\frac{4 a^3 b \cos ^9(c+d x)}{9 d}+\frac{4 a b^3 \cos ^9(c+d x)}{9 d}+\frac{a^4 \sin (c+d x)}{d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac{6 a^4 \sin ^5(c+d x)}{5 d}-\frac{18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{2 b^4 \sin ^7(c+d x)}{7 d}+\frac{a^4 \sin ^9(c+d x)}{9 d}-\frac{2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac{b^4 \sin ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.718227, size = 237, normalized size = 0.85 \[ \frac{1890 \left (14 a^2 b^2+21 a^4+b^4\right ) \sin (c+d x)+420 \left (21 a^4-b^4\right ) \sin (3 (c+d x))+252 \left (-12 a^2 b^2+9 a^4-b^4\right ) \sin (5 (c+d x))+45 \left (-30 a^2 b^2+9 a^4+b^4\right ) \sin (7 (c+d x))+35 \left (-6 a^2 b^2+a^4+b^4\right ) \sin (9 (c+d x))-2520 a b \left (7 a^2+3 b^2\right ) \cos (c+d x)-1680 a b \left (7 a^2+2 b^2\right ) \cos (3 (c+d x))-180 a b \left (7 a^2-3 b^2\right ) \cos (7 (c+d x))-140 a b \left (a^2-b^2\right ) \cos (9 (c+d x))-5040 a^3 b \cos (5 (c+d x))}{80640 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 236, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{9}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{21}}+{\frac{\sin \left ( dx+c \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) +4\,a{b}^{3} \left ( -1/9\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) +6\,{a}^{2}{b}^{2} \left ( -1/9\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{\sin \left ( dx+c \right ) }{63} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) } \right ) -{\frac{4\,{a}^{3}b \left ( \cos \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{{a}^{4}\sin \left ( dx+c \right ) }{9} \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56097, size = 251, normalized size = 0.9 \begin{align*} -\frac{140 \, a^{3} b \cos \left (d x + c\right )^{9} -{\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{4} + 6 \,{\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{2} b^{2} - 20 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a b^{3} -{\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} b^{4}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.55069, size = 413, normalized size = 1.48 \begin{align*} -\frac{180 \, a b^{3} \cos \left (d x + c\right )^{7} + 140 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{9} -{\left (35 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \,{\left (4 \, a^{4} + 3 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \,{\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 128 \, a^{4} + 96 \, a^{2} b^{2} + 8 \, b^{4} + 4 \,{\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.7535, size = 367, normalized size = 1.32 \begin{align*} \begin{cases} \frac{128 a^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac{64 a^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac{16 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac{8 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac{a^{4} \sin{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac{4 a^{3} b \cos ^{9}{\left (c + d x \right )}}{9 d} + \frac{32 a^{2} b^{2} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac{48 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac{12 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac{2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{8 a b^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac{8 b^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac{4 b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac{b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + b \sin{\left (c \right )}\right )^{4} \cos ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21216, size = 363, normalized size = 1.3 \begin{align*} -\frac{a^{3} b \cos \left (5 \, d x + 5 \, c\right )}{16 \, d} - \frac{{\left (a^{3} b - a b^{3}\right )} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} - \frac{{\left (7 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{{\left (7 \, a^{3} b + 2 \, a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac{{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )}{32 \, d} + \frac{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{{\left (9 \, a^{4} - 30 \, a^{2} b^{2} + b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac{{\left (9 \, a^{4} - 12 \, a^{2} b^{2} - b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (21 \, a^{4} - b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{3 \,{\left (21 \, a^{4} + 14 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{128 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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