Integrand size = 24, antiderivative size = 157 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=4 a \left (5 a^2+3 c^2\right ) x-\frac {4 c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{3 e}+\frac {4 a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{3 e}-\frac {20 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a+a \cos (d+e x)+c \sin (d+e x))}{3 e}-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))^2}{3 e} \]
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Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3199, 3225, 2717, 2718} \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=\frac {4 a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{3 e}-\frac {4 c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{3 e}+4 a x \left (5 a^2+3 c^2\right )-\frac {20 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a \cos (d+e x)+a+c \sin (d+e x))}{3 e}-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e} \]
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Rule 2717
Rule 2718
Rule 3199
Rule 3225
Rubi steps \begin{align*} \text {integral}& = -\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))^2}{3 e}+\frac {1}{3} \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x)) \left (4 \left (5 a^2+2 c^2\right )+20 a^2 \cos (d+e x)+20 a c \sin (d+e x)\right ) \, dx \\ & = -\frac {20 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a+a \cos (d+e x)+c \sin (d+e x))}{3 e}-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))^2}{3 e}+\frac {\int \left (48 a^2 \left (5 a^2+3 c^2\right )+16 a^2 \left (15 a^2+4 c^2\right ) \cos (d+e x)+16 a c \left (15 a^2+4 c^2\right ) \sin (d+e x)\right ) \, dx}{12 a} \\ & = 4 a \left (5 a^2+3 c^2\right ) x-\frac {20 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a+a \cos (d+e x)+c \sin (d+e x))}{3 e}-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))^2}{3 e}+\frac {1}{3} \left (4 a \left (15 a^2+4 c^2\right )\right ) \int \cos (d+e x) \, dx+\frac {1}{3} \left (4 c \left (15 a^2+4 c^2\right )\right ) \int \sin (d+e x) \, dx \\ & = 4 a \left (5 a^2+3 c^2\right ) x-\frac {4 c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{3 e}+\frac {4 a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{3 e}-\frac {20 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a+a \cos (d+e x)+c \sin (d+e x))}{3 e}-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))^2}{3 e} \\ \end{align*}
Time = 1.46 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.86 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=\frac {2 \left (6 a \left (5 a^2+3 c^2\right ) (d+e x)-9 c \left (5 a^2+c^2\right ) \cos (d+e x)-18 a^2 c \cos (2 (d+e x))+c \left (-3 a^2+c^2\right ) \cos (3 (d+e x))+9 a \left (5 a^2+c^2\right ) \sin (d+e x)+9 a \left (a^2-c^2\right ) \sin (2 (d+e x))+a \left (a^2-3 c^2\right ) \sin (3 (d+e x))\right )}{3 e} \]
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Time = 2.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(\frac {\left (-6 a^{2} c +2 c^{3}\right ) \cos \left (3 e x +3 d \right )+\left (18 a^{3}-18 a \,c^{2}\right ) \sin \left (2 e x +2 d \right )+\left (2 a^{3}-6 a \,c^{2}\right ) \sin \left (3 e x +3 d \right )-36 a^{2} c \cos \left (2 e x +2 d \right )+\left (-90 a^{2} c -18 c^{3}\right ) \cos \left (e x +d \right )+\left (90 a^{3}+18 a \,c^{2}\right ) \sin \left (e x +d \right )+60 a^{3} e x +36 a \,c^{2} e x -60 a^{2} c -16 c^{3}}{3 e}\) | \(152\) |
parts | \(\frac {-8 a^{2} c \cos \left (e x +d \right )^{3}+24 a^{3} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {8 a \left (a +c \sin \left (e x +d \right )\right )^{3}}{e c}+8 a^{3} x +\frac {8 a^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3 e}-\frac {8 c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3 e}-\frac {24 c \cos \left (e x +d \right ) a^{2}}{e}+\frac {24 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}\) | \(169\) |
derivativedivides | \(\frac {\frac {8 a^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-8 a^{2} c \cos \left (e x +d \right )^{3}+24 a^{3} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a \,c^{2} \sin \left (e x +d \right )^{3}-24 a^{2} c \cos \left (e x +d \right )^{2}+24 a^{3} \sin \left (e x +d \right )-\frac {8 c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}+24 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-24 a^{2} c \cos \left (e x +d \right )+8 a^{3} \left (e x +d \right )}{e}\) | \(177\) |
default | \(\frac {\frac {8 a^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-8 a^{2} c \cos \left (e x +d \right )^{3}+24 a^{3} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a \,c^{2} \sin \left (e x +d \right )^{3}-24 a^{2} c \cos \left (e x +d \right )^{2}+24 a^{3} \sin \left (e x +d \right )-\frac {8 c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}+24 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-24 a^{2} c \cos \left (e x +d \right )+8 a^{3} \left (e x +d \right )}{e}\) | \(177\) |
risch | \(20 a^{3} x +12 a \,c^{2} x -\frac {30 c \cos \left (e x +d \right ) a^{2}}{e}-\frac {6 c^{3} \cos \left (e x +d \right )}{e}+\frac {30 a^{3} \sin \left (e x +d \right )}{e}+\frac {6 a \sin \left (e x +d \right ) c^{2}}{e}-\frac {2 c \cos \left (3 e x +3 d \right ) a^{2}}{e}+\frac {2 c^{3} \cos \left (3 e x +3 d \right )}{3 e}+\frac {2 a^{3} \sin \left (3 e x +3 d \right )}{3 e}-\frac {2 a \sin \left (3 e x +3 d \right ) c^{2}}{e}-\frac {12 a^{2} c \cos \left (2 e x +2 d \right )}{e}+\frac {6 a^{3} \sin \left (2 e x +2 d \right )}{e}-\frac {6 a \sin \left (2 e x +2 d \right ) c^{2}}{e}\) | \(196\) |
norman | \(\frac {-\frac {192 a^{2} c +32 c^{3}}{3 e}+4 a \left (5 a^{2}+3 c^{2}\right ) x -\frac {32 c^{3} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e}+\frac {64 a \left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 e}+\frac {8 a \left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{e}+12 a \left (5 a^{2}+3 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+12 a \left (5 a^{2}+3 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+4 a \left (5 a^{2}+3 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}+\frac {8 a \left (11 a^{2}-3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(229\) |
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Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=-\frac {4 \, {\left (18 \, a^{2} c \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} e x + 6 \, {\left (3 \, a^{2} c + c^{3}\right )} \cos \left (e x + d\right ) - {\left (22 \, a^{3} + 6 \, a c^{2} + 2 \, {\left (a^{3} - 3 \, a c^{2}\right )} \cos \left (e x + d\right )^{2} + 9 \, {\left (a^{3} - a c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )}}{3 \, e} \]
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Time = 0.16 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=\begin {cases} 12 a^{3} x \sin ^{2}{\left (d + e x \right )} + 12 a^{3} x \cos ^{2}{\left (d + e x \right )} + 8 a^{3} x + \frac {16 a^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {8 a^{3} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} + \frac {12 a^{3} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {24 a^{3} \sin {\left (d + e x \right )}}{e} + \frac {24 a^{2} c \sin ^{2}{\left (d + e x \right )}}{e} - \frac {8 a^{2} c \cos ^{3}{\left (d + e x \right )}}{e} - \frac {24 a^{2} c \cos {\left (d + e x \right )}}{e} + 12 a c^{2} x \sin ^{2}{\left (d + e x \right )} + 12 a c^{2} x \cos ^{2}{\left (d + e x \right )} + \frac {8 a c^{2} \sin ^{3}{\left (d + e x \right )}}{e} - \frac {12 a c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {8 c^{3} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {16 c^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} & \text {for}\: e \neq 0 \\x \left (2 a \cos {\left (d \right )} + 2 a + 2 c \sin {\left (d \right )}\right )^{3} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=-\frac {8 \, a^{2} c \cos \left (e x + d\right )^{3}}{e} + \frac {8 \, a c^{2} \sin \left (e x + d\right )^{3}}{e} + 8 \, a^{3} x - \frac {8 \, {\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} a^{3}}{3 \, e} + \frac {8 \, {\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{3 \, e} - 24 \, a^{2} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {a \sin \left (e x + d\right )}{e}\right )} - 6 \, {\left (\frac {4 \, a c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} a^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} a \]
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Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=-\frac {12 \, a^{2} c \cos \left (2 \, e x + 2 \, d\right )}{e} + 4 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} x - \frac {2 \, {\left (3 \, a^{2} c - c^{3}\right )} \cos \left (3 \, e x + 3 \, d\right )}{3 \, e} - \frac {6 \, {\left (5 \, a^{2} c + c^{3}\right )} \cos \left (e x + d\right )}{e} + \frac {2 \, {\left (a^{3} - 3 \, a c^{2}\right )} \sin \left (3 \, e x + 3 \, d\right )}{3 \, e} + \frac {6 \, {\left (a^{3} - a c^{2}\right )} \sin \left (2 \, e x + 2 \, d\right )}{e} + \frac {6 \, {\left (5 \, a^{3} + a c^{2}\right )} \sin \left (e x + d\right )}{e} \]
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Time = 27.66 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.52 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=20\,a^3\,x-\frac {32\,c^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4}{e}+\frac {64\,c^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6}{3\,e}+12\,a\,c^2\,x-\frac {64\,a^2\,c\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6}{e}+\frac {40\,a^3\,\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e}+\frac {80\,a^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{3\,e}+\frac {64\,a^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{3\,e}+\frac {16\,a\,c^2\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e}-\frac {64\,a\,c^2\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e}+\frac {24\,a\,c^2\,\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e} \]
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