Integrand size = 20, antiderivative size = 170 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=\frac {1}{2} a \left (2 a^2+3 \left (b^2+c^2\right )\right ) x-\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)}{6 e}+\frac {b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 e}-\frac {5 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{6 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e} \]
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Time = 0.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3199, 3225, 2717, 2718} \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=\frac {b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 e}-\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)}{6 e}+\frac {1}{2} a x \left (2 a^2+3 \left (b^2+c^2\right )\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}-\frac {5 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{6 e} \]
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Rule 2717
Rule 2718
Rule 3199
Rule 3225
Rubi steps \begin{align*} \text {integral}& = -\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}+\frac {1}{3} \int (a+b \cos (d+e x)+c \sin (d+e x)) \left (3 a^2+2 \left (b^2+c^2\right )+5 a b \cos (d+e x)+5 a c \sin (d+e x)\right ) \, dx \\ & = -\frac {5 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{6 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}+\frac {\int \left (3 a^2 \left (2 a^2+3 \left (b^2+c^2\right )\right )+a b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)+a c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)\right ) \, dx}{6 a} \\ & = \frac {1}{2} a \left (2 a^2+3 \left (b^2+c^2\right )\right ) x-\frac {5 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{6 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}+\frac {1}{6} \left (b \left (11 a^2+4 \left (b^2+c^2\right )\right )\right ) \int \cos (d+e x) \, dx+\frac {1}{6} \left (c \left (11 a^2+4 \left (b^2+c^2\right )\right )\right ) \int \sin (d+e x) \, dx \\ & = \frac {1}{2} a \left (2 a^2+3 \left (b^2+c^2\right )\right ) x-\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)}{6 e}+\frac {b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 e}-\frac {5 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{6 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e} \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.85 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=\frac {6 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) (d+e x)-9 c \left (4 a^2+b^2+c^2\right ) \cos (d+e x)-18 a b c \cos (2 (d+e x))+c \left (-3 b^2+c^2\right ) \cos (3 (d+e x))+9 b \left (4 a^2+b^2+c^2\right ) \sin (d+e x)+9 a \left (b^2-c^2\right ) \sin (2 (d+e x))+b \left (b^2-3 c^2\right ) \sin (3 (d+e x))}{12 e} \]
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Time = 1.98 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99
method | result | size |
parts | \(a^{3} x +\frac {-b^{2} c \cos \left (e x +d \right )^{3}+3 a \,b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {b \left (a +c \sin \left (e x +d \right )\right )^{3}}{e c}+\frac {b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3 e}-\frac {c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3 e}-\frac {3 c \cos \left (e x +d \right ) a^{2}}{e}+\frac {3 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}\) | \(168\) |
derivativedivides | \(\frac {a^{3} \left (e x +d \right )+3 \sin \left (e x +d \right ) a^{2} b -3 a^{2} c \cos \left (e x +d \right )+3 a \,b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-3 a b c \cos \left (e x +d \right )^{2}+3 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+\frac {b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-b^{2} c \cos \left (e x +d \right )^{3}+c^{2} b \sin \left (e x +d \right )^{3}-\frac {c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}}{e}\) | \(177\) |
default | \(\frac {a^{3} \left (e x +d \right )+3 \sin \left (e x +d \right ) a^{2} b -3 a^{2} c \cos \left (e x +d \right )+3 a \,b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-3 a b c \cos \left (e x +d \right )^{2}+3 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+\frac {b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-b^{2} c \cos \left (e x +d \right )^{3}+c^{2} b \sin \left (e x +d \right )^{3}-\frac {c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}}{e}\) | \(177\) |
parallelrisch | \(\frac {\left (-3 b^{2} c +c^{3}\right ) \cos \left (3 e x +3 d \right )+9 a \left (b^{2}-c^{2}\right ) \sin \left (2 e x +2 d \right )+\left (b^{3}-3 c^{2} b \right ) \sin \left (3 e x +3 d \right )-18 a c b \cos \left (2 e x +2 d \right )+9 \left (-c^{3}+\left (-4 a^{2}-b^{2}\right ) c \right ) \cos \left (e x +d \right )+9 \left (4 a^{2} b +b^{3}+c^{2} b \right ) \sin \left (e x +d \right )-8 c^{3}+18 a \,c^{2} e x +6 \left (-6 a^{2}+3 a b -2 b^{2}\right ) c +12 e a \left (a^{2}+\frac {3 b^{2}}{2}\right ) x}{12 e}\) | \(177\) |
risch | \(a^{3} x +\frac {3 a \,b^{2} x}{2}+\frac {3 a \,c^{2} x}{2}-\frac {3 c \cos \left (e x +d \right ) a^{2}}{e}-\frac {3 c \cos \left (e x +d \right ) b^{2}}{4 e}-\frac {3 c^{3} \cos \left (e x +d \right )}{4 e}+\frac {3 b \sin \left (e x +d \right ) a^{2}}{e}+\frac {3 b^{3} \sin \left (e x +d \right )}{4 e}+\frac {3 b \sin \left (e x +d \right ) c^{2}}{4 e}-\frac {c \cos \left (3 e x +3 d \right ) b^{2}}{4 e}+\frac {c^{3} \cos \left (3 e x +3 d \right )}{12 e}+\frac {b^{3} \sin \left (3 e x +3 d \right )}{12 e}-\frac {b \sin \left (3 e x +3 d \right ) c^{2}}{4 e}-\frac {3 a c b \cos \left (2 e x +2 d \right )}{2 e}+\frac {3 a \sin \left (2 e x +2 d \right ) b^{2}}{4 e}-\frac {3 a \sin \left (2 e x +2 d \right ) c^{2}}{4 e}\) | \(232\) |
norman | \(\frac {\left (a^{3}+\frac {3}{2} a \,b^{2}+\frac {3}{2} a \,c^{2}\right ) x +\left (a^{3}+\frac {3}{2} a \,b^{2}+\frac {3}{2} a \,c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}+\left (3 a^{3}+\frac {9}{2} a \,b^{2}+\frac {9}{2} a \,c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\left (3 a^{3}+\frac {9}{2} a \,b^{2}+\frac {9}{2} a \,c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+\frac {\left (6 a^{2} b -3 a \,b^{2}+3 a \,c^{2}+2 b^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{e}+\frac {\left (6 a^{2} b +3 a \,b^{2}-3 a \,c^{2}+2 b^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}-\frac {18 a^{2} c +6 b^{2} c +4 c^{3}}{3 e}-\frac {\left (12 a^{2} c -12 a b c +4 c^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e}-\frac {3 \left (2 a^{2} c -4 a b c +2 b^{2} c \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{e}+\frac {4 b \left (9 a^{2}+b^{2}+6 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(324\) |
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Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.86 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=-\frac {18 \, a b c \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2} + 3 \, a c^{2}\right )} e x + 6 \, {\left (3 \, a^{2} c + c^{3}\right )} \cos \left (e x + d\right ) - {\left (18 \, a^{2} b + 4 \, b^{3} + 6 \, b c^{2} + 2 \, {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2} + 9 \, {\left (a b^{2} - a c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{6 \, e} \]
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Time = 0.16 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.73 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b \sin {\left (d + e x \right )}}{e} - \frac {3 a^{2} c \cos {\left (d + e x \right )}}{e} + \frac {3 a b^{2} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {3 a b^{2} x \cos ^{2}{\left (d + e x \right )}}{2} + \frac {3 a b^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} + \frac {3 a b c \sin ^{2}{\left (d + e x \right )}}{e} + \frac {3 a c^{2} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {3 a c^{2} x \cos ^{2}{\left (d + e x \right )}}{2} - \frac {3 a c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} + \frac {2 b^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {b^{3} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} - \frac {b^{2} c \cos ^{3}{\left (d + e x \right )}}{e} + \frac {b c^{2} \sin ^{3}{\left (d + e x \right )}}{e} - \frac {c^{3} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {2 c^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} & \text {for}\: e \neq 0 \\x \left (a + b \cos {\left (d \right )} + c \sin {\left (d \right )}\right )^{3} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.11 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=-\frac {b^{2} c \cos \left (e x + d\right )^{3}}{e} + \frac {b c^{2} \sin \left (e x + d\right )^{3}}{e} + a^{3} x - \frac {{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{3 \, e} + \frac {{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{3 \, e} - 3 \, a^{2} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} - \frac {3}{4} \, {\left (\frac {4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} a \]
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Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=-\frac {3 \, a b c \cos \left (2 \, e x + 2 \, d\right )}{2 \, e} + \frac {1}{2} \, {\left (2 \, a^{3} + 3 \, a b^{2} + 3 \, a c^{2}\right )} x - \frac {{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (3 \, e x + 3 \, d\right )}{12 \, e} - \frac {3 \, {\left (4 \, a^{2} c + b^{2} c + c^{3}\right )} \cos \left (e x + d\right )}{4 \, e} + \frac {{\left (b^{3} - 3 \, b c^{2}\right )} \sin \left (3 \, e x + 3 \, d\right )}{12 \, e} + \frac {3 \, {\left (a b^{2} - a c^{2}\right )} \sin \left (2 \, e x + 2 \, d\right )}{4 \, e} + \frac {3 \, {\left (4 \, a^{2} b + b^{3} + b c^{2}\right )} \sin \left (e x + d\right )}{4 \, e} \]
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Time = 29.07 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.96 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a^2+3\,b^2+3\,c^2\right )}{2\,a^3+3\,a\,b^2+3\,a\,c^2}\right )\,\left (2\,a^2+3\,b^2+3\,c^2\right )}{e}-\frac {a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )-\frac {e\,x}{2}\right )\,\left (2\,a^2+3\,b^2+3\,c^2\right )}{e}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (12\,a^2\,c-12\,b\,a\,c+4\,c^3\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (12\,a^2\,b+\frac {4\,b^3}{3}+8\,b\,c^2\right )-\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (6\,a^2\,b+3\,a\,b^2-3\,a\,c^2+2\,b^3\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (6\,c\,a^2-12\,c\,a\,b+6\,c\,b^2\right )+6\,a^2\,c+2\,b^2\,c-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (6\,a^2\,b-3\,a\,b^2+3\,a\,c^2+2\,b^3\right )+\frac {4\,c^3}{3}}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \]
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