Optimal. Leaf size=170 \[ \frac {\left (3 a^2+4 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \sin (c+d x) \cos (c+d x)}{30 d}+\frac {1}{2} a b x \left (4 a^2+3 b^2\right )+\frac {2 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \sin (c+d x)}{15 d}+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{5 d} \]
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Rubi [A] time = 0.20, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2753, 2734} \[ \frac {2 \left (28 a^2 b^2+3 a^4+4 b^4\right ) \sin (c+d x)}{15 d}+\frac {\left (3 a^2+4 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \sin (c+d x) \cos (c+d x)}{30 d}+\frac {1}{2} a b x \left (4 a^2+3 b^2\right )+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx &=\frac {(a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} \int (4 b+4 a \cos (c+d x)) (a+b \cos (c+d x))^3 \, dx\\ &=\frac {a (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {(a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{20} \int (a+b \cos (c+d x))^2 \left (28 a b+4 \left (3 a^2+4 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {\left (3 a^2+4 b^2\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac {a (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {(a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{60} \int (a+b \cos (c+d x)) \left (4 b \left (27 a^2+8 b^2\right )+4 a \left (6 a^2+29 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {1}{2} a b \left (4 a^2+3 b^2\right ) x+\frac {2 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \sin (c+d x)}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 a^2+4 b^2\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac {a (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {(a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 133, normalized size = 0.78 \[ \frac {30 \left (8 a^4+36 a^2 b^2+5 b^4\right ) \sin (c+d x)+b \left (480 a^3 c+480 a^3 d x+5 \left (24 a^2 b+5 b^3\right ) \sin (3 (c+d x))+240 a \left (a^2+b^2\right ) \sin (2 (c+d x))+30 a b^2 \sin (4 (c+d x))+360 a b^2 c+360 a b^2 d x+3 b^3 \sin (5 (c+d x))\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 121, normalized size = 0.71 \[ \frac {15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} d x + {\left (6 \, b^{4} \cos \left (d x + c\right )^{4} + 30 \, a b^{3} \cos \left (d x + c\right )^{3} + 30 \, a^{4} + 120 \, a^{2} b^{2} + 16 \, b^{4} + 4 \, {\left (15 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 134, normalized size = 0.79 \[ \frac {b^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {a b^{3} \sin \left (4 \, d x + 4 \, c\right )}{8 \, d} + \frac {1}{2} \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} x + \frac {{\left (24 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (a^{3} b + a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac {{\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 138, normalized size = 0.81 \[ \frac {\frac {b^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 133, normalized size = 0.78 \[ \frac {120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} b - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} b^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} + 8 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} b^{4} + 120 \, a^{4} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 363, normalized size = 2.14 \[ \frac {\left (2\,a^4-4\,a^3\,b+12\,a^2\,b^2-5\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (8\,a^4-8\,a^3\,b+32\,a^2\,b^2-2\,a\,b^3+\frac {8\,b^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,a^4+40\,a^2\,b^2+\frac {116\,b^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (8\,a^4+8\,a^3\,b+32\,a^2\,b^2+2\,a\,b^3+\frac {8\,b^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^4+4\,a^3\,b+12\,a^2\,b^2+5\,a\,b^3+2\,b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^2+3\,b^2\right )}{4\,a^3\,b+3\,a\,b^3}\right )\,\left (4\,a^2+3\,b^2\right )}{d}-\frac {a\,b\,\left (4\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.25, size = 301, normalized size = 1.77 \[ \begin {cases} \frac {a^{4} \sin {\left (c + d x \right )}}{d} + 2 a^{3} b x \sin ^{2}{\left (c + d x \right )} + 2 a^{3} b x \cos ^{2}{\left (c + d x \right )} + \frac {2 a^{3} b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {6 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a b^{3} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 a b^{3} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 a b^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 a b^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {8 b^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {b^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{4} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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