Optimal. Leaf size=275 \[ \frac{10 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{4 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \cos ^5(c+d x)}{d}-\frac{5 a^4 b \cos ^7(c+d x)}{7 d}-\frac{a^5 \sin ^7(c+d x)}{7 d}+\frac{3 a^5 \sin ^5(c+d x)}{5 d}-\frac{a^5 \sin ^3(c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a b^4 \sin ^7(c+d x)}{7 d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{b^5 \cos ^7(c+d x)}{7 d}+\frac{2 b^5 \cos ^5(c+d x)}{5 d}-\frac{b^5 \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.281352, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ \frac{10 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{4 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \cos ^5(c+d x)}{d}-\frac{5 a^4 b \cos ^7(c+d x)}{7 d}-\frac{a^5 \sin ^7(c+d x)}{7 d}+\frac{3 a^5 \sin ^5(c+d x)}{5 d}-\frac{a^5 \sin ^3(c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a b^4 \sin ^7(c+d x)}{7 d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{b^5 \cos ^7(c+d x)}{7 d}+\frac{2 b^5 \cos ^5(c+d x)}{5 d}-\frac{b^5 \cos ^3(c+d x)}{3 d} \]
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 ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos ^7(c+d x)+5 a^4 b \cos ^6(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^5(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^4(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^3(c+d x) \sin ^4(c+d x)+b^5 \cos ^2(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^5 \int \cos ^7(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^6(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^3(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^2(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac{a^5 \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{\left (5 a^4 b\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (10 a^3 b^2\right ) \operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (5 a b^4\right ) \operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac{b^5 \operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{5 a^4 b \cos ^7(c+d x)}{7 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{a^5 \sin ^3(c+d x)}{d}+\frac{3 a^5 \sin ^5(c+d x)}{5 d}-\frac{a^5 \sin ^7(c+d x)}{7 d}+\frac{\left (10 a^3 b^2\right ) \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (5 a b^4\right ) \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac{b^5 \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{b^5 \cos ^3(c+d x)}{3 d}-\frac{2 a^2 b^3 \cos ^5(c+d x)}{d}+\frac{2 b^5 \cos ^5(c+d x)}{5 d}-\frac{5 a^4 b \cos ^7(c+d x)}{7 d}+\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{b^5 \cos ^7(c+d x)}{7 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{a^5 \sin ^3(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{3 a^5 \sin ^5(c+d x)}{5 d}-\frac{4 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{a^5 \sin ^7(c+d x)}{7 d}+\frac{10 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{5 a b^4 \sin ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.7679, size = 236, normalized size = 0.86 \[ \frac{525 a \left (10 a^2 b^2+7 a^4+3 b^4\right ) \sin (c+d x)+35 a \left (-10 a^2 b^2+21 a^4-15 b^4\right ) \sin (3 (c+d x))+21 a \left (-30 a^2 b^2+7 a^4-5 b^4\right ) \sin (5 (c+d x))+15 a \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (7 (c+d x))-525 b \left (6 a^2 b^2+5 a^4+b^4\right ) \cos (c+d x)-35 b \left (30 a^2 b^2+45 a^4+b^4\right ) \cos (3 (c+d x))+21 b \left (10 a^2 b^2-25 a^4+3 b^4\right ) \cos (5 (c+d x))-15 b \left (-10 a^2 b^2+5 a^4+b^4\right ) \cos (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.201, size = 261, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{7}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{35}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{105}} \right ) +5\,a{b}^{4} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+1/35\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) +10\,{a}^{2}{b}^{3} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +10\,{a}^{3}{b}^{2} \left ( -1/7\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{5\,{a}^{4}b \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{{a}^{5}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08965, size = 262, normalized size = 0.95 \begin{align*} -\frac{75 \, a^{4} b \cos \left (d x + c\right )^{7} + 3 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{5} - 10 \,{\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{3} b^{2} - 30 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} b^{3} + 15 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a b^{4} +{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} b^{5}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.537526, size = 429, normalized size = 1.56 \begin{align*} -\frac{35 \, b^{5} \cos \left (d x + c\right )^{3} + 15 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{7} + 42 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{5} -{\left (15 \,{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{6} + 48 \, a^{5} + 80 \, a^{3} b^{2} + 30 \, a b^{4} + 6 \,{\left (3 \, a^{5} + 5 \, a^{3} b^{2} - 20 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} +{\left (24 \, a^{5} + 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.632, size = 357, normalized size = 1.3 \begin{align*} \begin{cases} \frac{16 a^{5} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 a^{5} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 a^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{a^{5} \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{5 a^{4} b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac{16 a^{3} b^{2} \sin ^{7}{\left (c + d x \right )}}{21 d} + \frac{8 a^{3} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{10 a^{3} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac{2 a^{2} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{d} - \frac{4 a^{2} b^{3} \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac{2 a b^{4} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac{a b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac{8 b^{5} \cos ^{7}{\left (c + d x \right )}}{105 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + b \sin{\left (c \right )}\right )^{5} \cos ^{2}{\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 = 1.3946, size = 350, normalized size = 1.27 \begin{align*} -\frac{{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{{\left (25 \, a^{4} b - 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{{\left (45 \, a^{4} b + 30 \, a^{2} b^{3} + b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{5 \,{\left (5 \, a^{4} b + 6 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )}{64 \, d} + \frac{{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{{\left (7 \, a^{5} - 30 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (21 \, a^{5} - 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{5 \,{\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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