Optimal. Leaf size=515 \[ \frac{a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac{a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35}{128} a^3 b^2 x-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac{9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{63 a^5 x}{256}-\frac{a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac{3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac{a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15}{256} a b^4 x+\frac{b^5 \sin ^{10}(c+d x)}{10 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.48451, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3090, 2635, 8, 2565, 30, 2568, 14, 2564, 266, 43} \[ \frac{a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac{a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35}{128} a^3 b^2 x-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac{9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{63 a^5 x}{256}-\frac{a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac{3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac{a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15}{256} a b^4 x+\frac{b^5 \sin ^{10}(c+d x)}{10 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 2635
Rule 8
Rule 2565
Rule 30
Rule 2568
Rule 14
Rule 2564
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos ^{10}(c+d x)+5 a^4 b \cos ^9(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^8(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^7(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^6(c+d x) \sin ^4(c+d x)+b^5 \cos ^5(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^5 \int \cos ^{10}(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^9(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^8(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^7(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^5(c+d x) \sin ^5(c+d x) \, dx\\ &=\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{1}{10} \left (9 a^5\right ) \int \cos ^8(c+d x) \, dx+\left (a^3 b^2\right ) \int \cos ^8(c+d x) \, dx+\frac{1}{2} \left (3 a b^4\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{\left (5 a^4 b\right ) \operatorname{Subst}\left (\int x^9 \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int x^7 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int x^5 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{1}{80} \left (63 a^5\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{8} \left (7 a^3 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{16} \left (3 a b^4\right ) \int \cos ^6(c+d x) \, dx-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (x^7-x^9\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int (1-x)^2 x^2 \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{1}{32} \left (21 a^5\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{48} \left (35 a^3 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{32} \left (5 a b^4\right ) \int \cos ^4(c+d x) \, dx+\frac{b^5 \operatorname{Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^{10}(c+d x)}{10 d}+\frac{1}{128} \left (63 a^5\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{64} \left (35 a^3 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{128} \left (15 a b^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^{10}(c+d x)}{10 d}+\frac{1}{256} \left (63 a^5\right ) \int 1 \, dx+\frac{1}{128} \left (35 a^3 b^2\right ) \int 1 \, dx+\frac{1}{256} \left (15 a b^4\right ) \int 1 \, dx\\ &=\frac{63 a^5 x}{256}+\frac{35}{128} a^3 b^2 x+\frac{15}{256} a b^4 x-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^{10}(c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 1.26564, size = 307, normalized size = 0.6 \[ \frac{120 a \left (70 a^2 b^2+63 a^4+15 b^4\right ) (c+d x)+300 a \left (14 a^2 b^2+21 a^4+b^4\right ) \sin (2 (c+d x))+600 a \left (-2 a^2 b^2+3 a^4-b^4\right ) \sin (4 (c+d x))+50 a \left (-26 a^2 b^2+9 a^4-3 b^4\right ) \sin (6 (c+d x))+75 a \left (-6 a^2 b^2+a^4+b^4\right ) \sin (8 (c+d x))+6 a \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (10 (c+d x))-1200 a^2 b \left (3 a^2+b^2\right ) \cos (4 (c+d x))-300 a^2 b \left (a^2-b^2\right ) \cos (8 (c+d x))-300 b \left (14 a^2 b^2+21 a^4+b^4\right ) \cos (2 (c+d x))+50 b \left (6 a^2 b^2-27 a^4+b^4\right ) \cos (6 (c+d x))-6 b \left (-10 a^2 b^2+5 a^4+b^4\right ) \cos (10 (c+d x))}{30720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.303, size = 335, normalized size = 0.7 \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 ) ^{6}}{10}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{60}} \right ) +5\,a{b}^{4} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +10\,{a}^{2}{b}^{3} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}-1/40\, \left ( \cos \left ( dx+c \right ) \right ) ^{8} \right ) +10\,{a}^{3}{b}^{2} \left ( -1/10\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{\sin \left ( dx+c \right ) }{80} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+7/6\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{7\,dx}{256}}+{\frac{7\,c}{256}} \right ) -{\frac{{a}^{4}b \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{2}}+{a}^{5} \left ({\frac{\sin \left ( dx+c \right ) }{10} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{21\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{64}}+{\frac{315\,\cos \left ( dx+c \right ) }{128}} \right ) }+{\frac{63\,dx}{256}}+{\frac{63\,c}{256}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23067, size = 392, normalized size = 0.76 \begin{align*} -\frac{15360 \, a^{4} b \cos \left (d x + c\right )^{10} - 3 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 2520 \, d x + 2520 \, c + 25 \, \sin \left (8 \, d x + 8 \, c\right ) + 600 \, \sin \left (4 \, d x + 4 \, c\right ) + 2560 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} + 10 \,{\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c + 45 \, \sin \left (8 \, d x + 8 \, c\right ) + 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} + 7680 \,{\left (4 \, \sin \left (d x + c\right )^{10} - 15 \, \sin \left (d x + c\right )^{8} + 20 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} - 512 \,{\left (6 \, \sin \left (d x + c\right )^{10} - 15 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{30720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.605827, size = 595, normalized size = 1.16 \begin{align*} -\frac{640 \, b^{5} \cos \left (d x + c\right )^{6} + 384 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{10} + 960 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{8} - 15 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x -{\left (384 \,{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 48 \,{\left (9 \, a^{5} + 10 \, a^{3} b^{2} - 55 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 48.4455, size = 979, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37491, size = 462, normalized size = 0.9 \begin{align*} \frac{1}{256} \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x - \frac{{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac{5 \,{\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac{5 \,{\left (27 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{5 \,{\left (3 \, a^{4} b + a^{2} b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac{5 \,{\left (21 \, a^{4} b + 14 \, a^{2} b^{3} + b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac{{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{5 \,{\left (a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac{5 \,{\left (9 \, a^{5} - 26 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} + \frac{5 \,{\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{5 \,{\left (21 \, a^{5} + 14 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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