Optimal. Leaf size=150 \[ -\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{7}{8} a^4 x (5 A+4 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
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Rubi [A] time = 0.139084, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{7}{8} a^4 x (5 A+4 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2645
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx &=\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} (5 A+4 B) \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} (5 A+4 B) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx\\ &=\frac{1}{5} a^4 (5 A+4 B) x+\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} \left (a^4 (5 A+4 B)\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{5} \left (4 a^4 (5 A+4 B)\right ) \int \cos (c+d x) \, dx+\frac{1}{5} \left (4 a^4 (5 A+4 B)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{5} \left (6 a^4 (5 A+4 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{5} a^4 (5 A+4 B) x+\frac{4 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{3 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{20} \left (3 a^4 (5 A+4 B)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{5} \left (3 a^4 (5 A+4 B)\right ) \int 1 \, dx-\frac{\left (4 a^4 (5 A+4 B)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{4}{5} a^4 (5 A+4 B) x+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{27 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{40 d}+\frac{a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{1}{40} \left (3 a^4 (5 A+4 B)\right ) \int 1 \, dx\\ &=\frac{7}{8} a^4 (5 A+4 B) x+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{27 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{40 d}+\frac{a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.335601, size = 108, normalized size = 0.72 \[ \frac{a^4 (420 (8 A+7 B) \sin (c+d x)+120 (7 A+8 B) \sin (2 (c+d x))+160 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+2100 A d x+290 B \sin (3 (c+d x))+60 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+1680 B d x)}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 248, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4}B\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 ) }+A{a}^{4} \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 ) +4\,{a}^{4}B \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{4\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +6\,A{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,{a}^{4}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,A{a}^{4}\sin \left ( dx+c \right ) +{a}^{4}B\sin \left ( dx+c \right ) +A{a}^{4} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0301, size = 319, normalized size = 2.13 \begin{align*} -\frac{640 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 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^{4} - 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 480 \,{\left (d x + c\right )} A a^{4} - 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 )} B a^{4} + 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 60 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 480 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1920 \, A a^{4} \sin \left (d x + c\right ) - 480 \, B a^{4} \sin \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31348, size = 279, normalized size = 1.86 \begin{align*} \frac{105 \,{\left (5 \, A + 4 \, B\right )} a^{4} d x +{\left (24 \, B a^{4} \cos \left (d x + c\right )^{4} + 30 \,{\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (10 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \,{\left (27 \, A + 28 \, B\right )} a^{4} \cos \left (d x + c\right ) + 8 \,{\left (100 \, A + 83 \, B\right )} a^{4}\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 [A] time = 4.46919, size = 544, normalized size = 3.63 \begin{align*} \begin{cases} \frac{3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac{3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 A a^{4} x \cos ^{2}{\left (c + d x \right )} + A a^{4} x + \frac{3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{8 A a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 A a^{4} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{4 A a^{4} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 A a^{4} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{4 A a^{4} \sin{\left (c + d x \right )}}{d} + \frac{3 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 B a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac{3 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 B a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac{8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{4 B a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{B a^{4} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{5 B a^{4} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac{6 B a^{4} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{2 B a^{4} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{B a^{4} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right )^{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23771, size = 188, normalized size = 1.25 \begin{align*} \frac{B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{7}{8} \,{\left (5 \, A a^{4} + 4 \, B a^{4}\right )} x + \frac{{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (16 \, A a^{4} + 29 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{7 \,{\left (8 \, A a^{4} + 7 \, B a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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