Optimal. Leaf size=220 \[ \frac{6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^7(c+d x)}{7 d}-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{3 a^4 \sin ^5(c+d x)}{5 d}-\frac{a^4 \sin ^3(c+d x)}{d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^7(c+d x)}{7 d}-\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.234498, antiderivative size = 220, 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} \[ \frac{6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^7(c+d x)}{7 d}-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{3 a^4 \sin ^5(c+d x)}{5 d}-\frac{a^4 \sin ^3(c+d x)}{d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^7(c+d x)}{7 d}-\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d} \]
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 ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \cos ^7(c+d x)+4 a^3 b \cos ^6(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^5(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^4(c+d x) \sin ^3(c+d x)+b^4 \cos ^3(c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^7(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^6(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^3(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac{a^4 \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 (4 a^3 b\right ) \operatorname{Subst}\left (\int x^6 \, 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 )^2 \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int x^4 \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 ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{4 a^3 b \cos ^7(c+d x)}{7 d}+\frac{a^4 \sin (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x)}{d}+\frac{3 a^4 \sin ^5(c+d x)}{5 d}-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{\left (6 a^2 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 (4 a b^3\right ) \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^4 \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{4 a^3 b \cos ^7(c+d x)}{7 d}+\frac{4 a b^3 \cos ^7(c+d x)}{7 d}+\frac{a^4 \sin (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x)}{d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac{3 a^4 \sin ^5(c+d x)}{5 d}-\frac{12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{b^4 \sin ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.520488, size = 204, normalized size = 0.93 \[ \frac{35 \left (30 a^2 b^2+35 a^4+3 b^4\right ) \sin (c+d x)+35 \left (-2 a^2 b^2+7 a^4-b^4\right ) \sin (3 (c+d x))+7 \left (-18 a^2 b^2+7 a^4-b^4\right ) \sin (5 (c+d x))+5 \left (-6 a^2 b^2+a^4+b^4\right ) \sin (7 (c+d x))-140 a b \left (5 a^2+3 b^2\right ) \cos (c+d x)-140 a b \left (3 a^2+b^2\right ) \cos (3 (c+d x))-28 a b \left (5 a^2-b^2\right ) \cos (5 (c+d x))-20 a b \left (a^2-b^2\right ) \cos (7 (c+d x))}{2240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 206, 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 ) ^{4}}{7}}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{ \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{35}} \right ) +4\,a{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 ) +6\,{a}^{2}{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{4\,{a}^{3}b \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{{a}^{4}\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 = 1.14551, size = 208, normalized size = 0.95 \begin{align*} -\frac{20 \, a^{3} b \cos \left (d x + c\right )^{7} +{\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^{4} - 2 \,{\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^{2} b^{2} - 4 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a b^{3} +{\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} b^{4}}{35 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.525996, size = 336, normalized size = 1.53 \begin{align*} -\frac{28 \, a b^{3} \cos \left (d x + c\right )^{5} + 20 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{7} -{\left (5 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 2 \,{\left (3 \, a^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, a^{4} + 16 \, a^{2} b^{2} + 2 \, b^{4} +{\left (8 \, a^{4} + 8 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.20488, size = 286, normalized size = 1.3 \begin{align*} \begin{cases} \frac{16 a^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{a^{4} \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{4 a^{3} b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac{16 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{8 a b^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac{2 b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\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 ^{3}{\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 = 1.21194, size = 309, normalized size = 1.4 \begin{align*} -\frac{{\left (a^{3} b - a b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{112 \, d} - \frac{{\left (5 \, a^{3} b - a b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac{{\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac{{\left (5 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )}{16 \, d} + \frac{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{{\left (7 \, a^{4} - 18 \, a^{2} b^{2} - b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (7 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac{{\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, 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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