Integrand size = 24, antiderivative size = 157 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=4 a \left (5 a^2+3 b^2\right ) x+\frac {4 a \left (15 a^2+4 b^2\right ) \cos (d+e x)}{3 e}+\frac {4 b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{3 e}+\frac {8 (a+b \cos (d+e x)-a \sin (d+e x))^2 (a \cos (d+e x)+b \sin (d+e x))}{3 e}+\frac {20 (a+b \cos (d+e x)-a \sin (d+e x)) \left (a^2 \cos (d+e x)+a b \sin (d+e x)\right )}{3 e} \]
[Out]
Time = 0.16 (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} \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\frac {4 b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{3 e}+\frac {4 a \left (15 a^2+4 b^2\right ) \cos (d+e x)}{3 e}+4 a x \left (5 a^2+3 b^2\right )+\frac {20 \left (a^2 \cos (d+e x)+a b \sin (d+e x)\right ) (a (-\sin (d+e x))+a+b \cos (d+e x))}{3 e}+\frac {8 (a \cos (d+e x)+b \sin (d+e x)) (a (-\sin (d+e x))+a+b \cos (d+e x))^2}{3 e} \]
[In]
[Out]
Rule 2717
Rule 2718
Rule 3199
Rule 3225
Rubi steps \begin{align*} \text {integral}& = \frac {8 (a+b \cos (d+e x)-a \sin (d+e x))^2 (a \cos (d+e x)+b \sin (d+e x))}{3 e}+\frac {1}{3} \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \left (4 \left (5 a^2+2 b^2\right )+20 a b \cos (d+e x)-20 a^2 \sin (d+e x)\right ) \, dx \\ & = \frac {8 (a+b \cos (d+e x)-a \sin (d+e x))^2 (a \cos (d+e x)+b \sin (d+e x))}{3 e}+\frac {20 (a+b \cos (d+e x)-a \sin (d+e x)) \left (a^2 \cos (d+e x)+a b \sin (d+e x)\right )}{3 e}+\frac {\int \left (48 a^2 \left (5 a^2+3 b^2\right )+16 a b \left (15 a^2+4 b^2\right ) \cos (d+e x)-16 a^2 \left (15 a^2+4 b^2\right ) \sin (d+e x)\right ) \, dx}{12 a} \\ & = 4 a \left (5 a^2+3 b^2\right ) x+\frac {8 (a+b \cos (d+e x)-a \sin (d+e x))^2 (a \cos (d+e x)+b \sin (d+e x))}{3 e}+\frac {20 (a+b \cos (d+e x)-a \sin (d+e x)) \left (a^2 \cos (d+e x)+a b \sin (d+e x)\right )}{3 e}-\frac {1}{3} \left (4 a \left (15 a^2+4 b^2\right )\right ) \int \sin (d+e x) \, dx+\frac {1}{3} \left (4 b \left (15 a^2+4 b^2\right )\right ) \int \cos (d+e x) \, dx \\ & = 4 a \left (5 a^2+3 b^2\right ) x+\frac {4 a \left (15 a^2+4 b^2\right ) \cos (d+e x)}{3 e}+\frac {4 b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{3 e}+\frac {8 (a+b \cos (d+e x)-a \sin (d+e x))^2 (a \cos (d+e x)+b \sin (d+e x))}{3 e}+\frac {20 (a+b \cos (d+e x)-a \sin (d+e x)) \left (a^2 \cos (d+e x)+a b \sin (d+e x)\right )}{3 e} \\ \end{align*}
Time = 1.67 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\frac {2 \left (6 a \left (5 a^2+3 b^2\right ) (d+e x)+9 a \left (5 a^2+b^2\right ) \cos (d+e x)+18 a^2 b \cos (2 (d+e x))-a \left (a^2-3 b^2\right ) \cos (3 (d+e x))+9 b \left (5 a^2+b^2\right ) \sin (d+e x)-9 a \left (a^2-b^2\right ) \sin (2 (d+e x))+b \left (-3 a^2+b^2\right ) \sin (3 (d+e x))\right )}{3 e} \]
[In]
[Out]
Time = 2.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(\frac {\frac {2 \left (-a^{3}+3 a \,b^{2}\right ) \cos \left (3 e x +3 d \right )}{3}+6 \left (-a^{3}+a \,b^{2}\right ) \sin \left (2 e x +2 d \right )+\frac {2 \left (-3 a^{2} b +b^{3}\right ) \sin \left (3 e x +3 d \right )}{3}+12 a^{2} b \cos \left (2 e x +2 d \right )+6 \left (5 a^{3}+a \,b^{2}\right ) \cos \left (e x +d \right )+6 \left (5 a^{2} b +b^{3}\right ) \sin \left (e x +d \right )+\frac {4 \left (15 e x +22\right ) a^{3}}{3}-12 a^{2} b +4 \left (3 e x +2\right ) b^{2} a}{e}\) | \(152\) |
parts | \(\frac {8 a \,b^{2} \cos \left (e x +d \right )^{3}+24 a \,b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {8 a^{2} b \left (\sin \left (e x +d \right )-1\right )^{3}}{e}+8 a^{3} x +\frac {24 a^{3} \cos \left (e x +d \right )}{e}+\frac {24 a^{3} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {8 b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3 e}+\frac {8 a^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3 e}\) | \(166\) |
derivativedivides | \(\frac {\frac {8 b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}+8 a \,b^{2} \cos \left (e x +d \right )^{3}+24 a \,b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a^{2} b \sin \left (e x +d \right )^{3}+24 a^{2} b \cos \left (e x +d \right )^{2}+24 \sin \left (e x +d \right ) a^{2} b +\frac {8 a^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}+24 a^{3} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+24 a^{3} \cos \left (e x +d \right )+8 a^{3} \left (e x +d \right )}{e}\) | \(176\) |
default | \(\frac {\frac {8 b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}+8 a \,b^{2} \cos \left (e x +d \right )^{3}+24 a \,b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a^{2} b \sin \left (e x +d \right )^{3}+24 a^{2} b \cos \left (e x +d \right )^{2}+24 \sin \left (e x +d \right ) a^{2} b +\frac {8 a^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}+24 a^{3} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+24 a^{3} \cos \left (e x +d \right )+8 a^{3} \left (e x +d \right )}{e}\) | \(176\) |
risch | \(20 a^{3} x +12 a \,b^{2} x +\frac {30 a^{3} \cos \left (e x +d \right )}{e}+\frac {6 a \cos \left (e x +d \right ) b^{2}}{e}+\frac {30 b \sin \left (e x +d \right ) a^{2}}{e}+\frac {6 b^{3} \sin \left (e x +d \right )}{e}-\frac {2 a^{3} \cos \left (3 e x +3 d \right )}{3 e}+\frac {2 a \cos \left (3 e x +3 d \right ) b^{2}}{e}-\frac {2 b \sin \left (3 e x +3 d \right ) a^{2}}{e}+\frac {2 b^{3} \sin \left (3 e x +3 d \right )}{3 e}+\frac {12 a^{2} b \cos \left (2 e x +2 d \right )}{e}-\frac {6 a^{3} \sin \left (2 e x +2 d \right )}{e}+\frac {6 a \sin \left (2 e x +2 d \right ) b^{2}}{e}\) | \(196\) |
norman | \(\frac {\left (20 a^{3}+12 a \,b^{2}\right ) x +\left (20 a^{3}+12 a \,b^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}+\left (60 a^{3}+36 a \,b^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\left (60 a^{3}+36 a \,b^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+\frac {\left (128 a^{3}-96 a^{2} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e}+\frac {176 a^{3}+48 a \,b^{2}}{3 e}-\frac {8 \left (3 a^{3}-6 a^{2} b -3 a \,b^{2}-2 b^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {8 \left (3 a^{3}+6 a^{2} b -3 a \,b^{2}+2 b^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{e}+\frac {3 \left (16 a^{3}-32 a^{2} b +16 a \,b^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{e}+\frac {32 b \left (15 a^{2}+b^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(287\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.80 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\frac {4 \, {\left (18 \, a^{2} b \cos \left (e x + d\right )^{2} + 24 \, a^{3} \cos \left (e x + d\right ) - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} e x + {\left (24 \, a^{2} b + 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (e x + d\right )^{2} - 9 \, {\left (a^{3} - a b^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )}}{3 \, e} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\begin {cases} 12 a^{3} x \sin ^{2}{\left (d + e x \right )} + 12 a^{3} x \cos ^{2}{\left (d + e x \right )} + 8 a^{3} x + \frac {8 a^{3} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {12 a^{3} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {16 a^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} + \frac {24 a^{3} \cos {\left (d + e x \right )}}{e} + \frac {8 a^{2} b \sin ^{3}{\left (d + e x \right )}}{e} - \frac {24 a^{2} b \sin ^{2}{\left (d + e x \right )}}{e} + \frac {24 a^{2} b \sin {\left (d + e x \right )}}{e} + 12 a b^{2} x \sin ^{2}{\left (d + e x \right )} + 12 a b^{2} x \cos ^{2}{\left (d + e x \right )} + \frac {12 a b^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {8 a b^{2} \cos ^{3}{\left (d + e x \right )}}{e} + \frac {16 b^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {8 b^{3} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \left (- 2 a \sin {\left (d \right )} + 2 a + 2 b \cos {\left (d \right )}\right )^{3} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.20 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\frac {8 \, a b^{2} \cos \left (e x + d\right )^{3}}{e} + \frac {8 \, a^{2} b \sin \left (e x + d\right )^{3}}{e} + 8 \, a^{3} x - \frac {8 \, {\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} a^{3}}{3 \, e} - \frac {8 \, {\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{3 \, e} + 24 \, a^{2} {\left (\frac {a \cos \left (e x + d\right )}{e} + \frac {b \sin \left (e x + d\right )}{e}\right )} + 6 \, {\left (\frac {4 \, a b \cos \left (e x + d\right )^{2}}{e} + \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} a^{2}}{e} + \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e}\right )} a \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\frac {12 \, a^{2} b \cos \left (2 \, e x + 2 \, d\right )}{e} + 4 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (3 \, e x + 3 \, d\right )}{3 \, e} + \frac {6 \, {\left (5 \, a^{3} + a b^{2}\right )} \cos \left (e x + d\right )}{e} - \frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \sin \left (3 \, e x + 3 \, d\right )}{3 \, e} - \frac {6 \, {\left (a^{3} - a b^{2}\right )} \sin \left (2 \, e x + 2 \, d\right )}{e} + \frac {6 \, {\left (5 \, a^{2} b + b^{3}\right )} \sin \left (e x + d\right )}{e} \]
[In]
[Out]
Time = 27.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.86 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (48\,a^3-96\,a^2\,b+48\,a\,b^2\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (24\,a^3+48\,a^2\,b-24\,a\,b^2+16\,b^3\right )+16\,a\,b^2-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (96\,a^2\,b-128\,a^3\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (160\,a^2\,b+\frac {32\,b^3}{3}\right )+\frac {176\,a^3}{3}+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (-24\,a^3+48\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )}+\frac {8\,a\,\mathrm {atan}\left (\frac {8\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (5\,a^2+3\,b^2\right )}{40\,a^3+24\,a\,b^2}\right )\,\left (5\,a^2+3\,b^2\right )}{e}-\frac {8\,a\,\left (5\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )-\frac {e\,x}{2}\right )}{e} \]
[In]
[Out]