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 = 0.14, antiderivative size = 150, normalized size of antiderivative = 1.00, 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 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
Rule 2751
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*}
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Mathematica [A] time = 0.38, 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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fricas [A] time = 0.62, size = 110, normalized size = 0.73 \[ \frac {105 \, {\left (5 \, A + 4 \, B\right )} a^{4} d x + {\left (24 \, B a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (10 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (27 \, A + 28 \, B\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (100 \, A + 83 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 139, normalized size = 0.93 \[ \frac {B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {7}{8} \, {\left (5 \, A a^{4} + 4 \, B a^{4}\right )} x + \frac {{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (16 \, A a^{4} + 29 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {7 \, {\left (8 \, A a^{4} + 7 \, B a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 248, normalized size = 1.65 \[ \frac {\frac {a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{4} \sin \left (d x +c \right )+a^{4} B \sin \left (d x +c \right )+A \,a^{4} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 236, normalized size = 1.57 \[ -\frac {640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 480 \, {\left (d x + c\right )} A a^{4} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1920 \, A a^{4} \sin \left (d x + c\right ) - 480 \, B a^{4} \sin \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 278, normalized size = 1.85 \[ \frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {245\,A\,a^4}{6}+\frac {98\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {224\,A\,a^4}{3}+\frac {896\,B\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {395\,A\,a^4}{6}+\frac {158\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+25\,B\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^4\,\left (5\,A+4\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A+4\,B\right )}{4\,\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4\right )}\right )\,\left (5\,A+4\,B\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.02, size = 544, normalized size = 3.63 \[ \begin {cases} \frac {3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 A a^{4} x \cos ^{2}{\left (c + d x \right )} + A a^{4} x + \frac {3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {8 A a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 A a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {4 A a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 A a^{4} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 B a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 B a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 B a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 B a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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