Integrand size = 14, antiderivative size = 36 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {3 b x}{8}-\frac {3}{8} b \cos (x) \sin (x)-\frac {1}{4} b \cos (x) \sin ^3(x)+\frac {1}{4} a \sin ^4(x) \] Output:
3/8*b*x-3/8*b*cos(x)*sin(x)-1/4*b*cos(x)*sin(x)^3+1/4*a*sin(x)^4
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {3 b x}{8}+\frac {1}{4} a \sin ^4(x)-\frac {1}{4} b \sin (2 x)+\frac {1}{32} b \sin (4 x) \] Input:
Integrate[Sin[x]^3*(a*Cos[x] + b*Sin[x]),x]
Output:
(3*b*x)/8 + (a*Sin[x]^4)/4 - (b*Sin[2*x])/4 + (b*Sin[4*x])/32
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 3568, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x)^3 (a \cos (x)+b \sin (x))dx\) |
\(\Big \downarrow \) 3568 |
\(\displaystyle \int \left (a \sin ^3(x) \cos (x)+b \sin ^4(x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} a \sin ^4(x)+\frac {3 b x}{8}-\frac {1}{4} b \sin ^3(x) \cos (x)-\frac {3}{8} b \sin (x) \cos (x)\) |
Input:
Int[Sin[x]^3*(a*Cos[x] + b*Sin[x]),x]
Output:
(3*b*x)/8 - (3*b*Cos[x]*Sin[x])/8 - (b*Cos[x]*Sin[x]^3)/4 + (a*Sin[x]^4)/4
Int[sin[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[sin[c + d*x]^m*(a *cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte gerQ[m] && IGtQ[n, 0]
Time = 0.57 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78
method | result | size |
default | \(b \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )+\frac {a \sin \left (x \right )^{4}}{4}\) | \(28\) |
parts | \(b \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )+\frac {a \sin \left (x \right )^{4}}{4}\) | \(28\) |
risch | \(\frac {3 b x}{8}+\frac {a \cos \left (4 x \right )}{32}+\frac {b \sin \left (4 x \right )}{32}-\frac {a \cos \left (2 x \right )}{8}-\frac {b \sin \left (2 x \right )}{4}\) | \(34\) |
parallelrisch | \(\frac {3 b x}{8}+\frac {3 a}{32}-\frac {a \cos \left (2 x \right )}{8}+\frac {a \cos \left (4 x \right )}{32}+\frac {b \sin \left (4 x \right )}{32}-\frac {b \sin \left (2 x \right )}{4}\) | \(37\) |
norman | \(\frac {4 a \tan \left (\frac {x}{2}\right )^{4}+\frac {3 b x}{8}-\frac {3 b \tan \left (\frac {x}{2}\right )}{4}-\frac {11 b \tan \left (\frac {x}{2}\right )^{3}}{4}+\frac {11 b \tan \left (\frac {x}{2}\right )^{5}}{4}+\frac {3 b \tan \left (\frac {x}{2}\right )^{7}}{4}+\frac {3 b x \tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {9 b x \tan \left (\frac {x}{2}\right )^{4}}{4}+\frac {3 b x \tan \left (\frac {x}{2}\right )^{6}}{2}+\frac {3 b x \tan \left (\frac {x}{2}\right )^{8}}{8}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}\) | \(100\) |
orering | \(x \sin \left (x \right )^{3} \left (a \cos \left (x \right )+b \sin \left (x \right )\right )-\frac {39 \sin \left (x \right )^{2} \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \cos \left (x \right )}{64}-\frac {11 \sin \left (x \right )^{3} \left (b \cos \left (x \right )-a \sin \left (x \right )\right )}{64}+\frac {5 x \left (6 \sin \left (x \right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \cos \left (x \right )^{2}+6 \sin \left (x \right )^{2} \left (b \cos \left (x \right )-a \sin \left (x \right )\right ) \cos \left (x \right )-3 \sin \left (x \right )^{3} \left (a \cos \left (x \right )+b \sin \left (x \right )\right )+\sin \left (x \right )^{3} \left (-b \sin \left (x \right )-a \cos \left (x \right )\right )\right )}{16}-\frac {3 \cos \left (x \right )^{3} \left (a \cos \left (x \right )+b \sin \left (x \right )\right )}{32}-\frac {9 \sin \left (x \right ) \left (b \cos \left (x \right )-a \sin \left (x \right )\right ) \cos \left (x \right )^{2}}{32}-\frac {9 \sin \left (x \right )^{2} \left (-b \sin \left (x \right )-a \cos \left (x \right )\right ) \cos \left (x \right )}{64}-\frac {\sin \left (x \right )^{3} \left (-b \cos \left (x \right )+a \sin \left (x \right )\right )}{64}+\frac {x \left (-60 \sin \left (x \right ) \left (a \cos \left (x \right )+b \sin \left (x \right )\right ) \cos \left (x \right )^{2}+24 \cos \left (x \right )^{3} \left (b \cos \left (x \right )-a \sin \left (x \right )\right )+36 \sin \left (x \right ) \left (-b \sin \left (x \right )-a \cos \left (x \right )\right ) \cos \left (x \right )^{2}-84 \sin \left (x \right )^{2} \left (b \cos \left (x \right )-a \sin \left (x \right )\right ) \cos \left (x \right )+22 \sin \left (x \right )^{3} \left (a \cos \left (x \right )+b \sin \left (x \right )\right )+12 \sin \left (x \right )^{2} \left (-b \cos \left (x \right )+a \sin \left (x \right )\right ) \cos \left (x \right )-18 \sin \left (x \right )^{3} \left (-b \sin \left (x \right )-a \cos \left (x \right )\right )\right )}{64}\) | \(312\) |
Input:
int(sin(x)^3*(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)
Output:
b*(-1/4*(sin(x)^3+3/2*sin(x))*cos(x)+3/8*x)+1/4*a*sin(x)^4
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {1}{4} \, a \cos \left (x\right )^{4} - \frac {1}{2} \, a \cos \left (x\right )^{2} + \frac {3}{8} \, b x + \frac {1}{8} \, {\left (2 \, b \cos \left (x\right )^{3} - 5 \, b \cos \left (x\right )\right )} \sin \left (x\right ) \] Input:
integrate(sin(x)^3*(a*cos(x)+b*sin(x)),x, algorithm="fricas")
Output:
1/4*a*cos(x)^4 - 1/2*a*cos(x)^2 + 3/8*b*x + 1/8*(2*b*cos(x)^3 - 5*b*cos(x) )*sin(x)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (37) = 74\).
Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.08 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {a \sin ^{4}{\left (x \right )}}{4} + \frac {3 b x \sin ^{4}{\left (x \right )}}{8} + \frac {3 b x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac {3 b x \cos ^{4}{\left (x \right )}}{8} - \frac {5 b \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{8} - \frac {3 b \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{8} \] Input:
integrate(sin(x)**3*(a*cos(x)+b*sin(x)),x)
Output:
a*sin(x)**4/4 + 3*b*x*sin(x)**4/8 + 3*b*x*sin(x)**2*cos(x)**2/4 + 3*b*x*co s(x)**4/8 - 5*b*sin(x)**3*cos(x)/8 - 3*b*sin(x)*cos(x)**3/8
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {1}{4} \, a \sin \left (x\right )^{4} + \frac {1}{32} \, b {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} \] Input:
integrate(sin(x)^3*(a*cos(x)+b*sin(x)),x, algorithm="maxima")
Output:
1/4*a*sin(x)^4 + 1/32*b*(12*x + sin(4*x) - 8*sin(2*x))
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {3}{8} \, b x + \frac {1}{32} \, a \cos \left (4 \, x\right ) - \frac {1}{8} \, a \cos \left (2 \, x\right ) + \frac {1}{32} \, b \sin \left (4 \, x\right ) - \frac {1}{4} \, b \sin \left (2 \, x\right ) \] Input:
integrate(sin(x)^3*(a*cos(x)+b*sin(x)),x, algorithm="giac")
Output:
3/8*b*x + 1/32*a*cos(4*x) - 1/8*a*cos(2*x) + 1/32*b*sin(4*x) - 1/4*b*sin(2 *x)
Time = 16.84 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {a\,{\cos \left (x\right )}^4}{4}+\frac {b\,\sin \left (x\right )\,{\cos \left (x\right )}^3}{4}-\frac {a\,{\cos \left (x\right )}^2}{2}-\frac {5\,b\,\sin \left (x\right )\,\cos \left (x\right )}{8}+\frac {3\,b\,x}{8} \] Input:
int(sin(x)^3*(a*cos(x) + b*sin(x)),x)
Output:
(3*b*x)/8 - (a*cos(x)^2)/2 + (a*cos(x)^4)/4 - (5*b*cos(x)*sin(x))/8 + (b*c os(x)^3*sin(x))/4
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx=-\frac {\cos \left (x \right ) \sin \left (x \right )^{3} b}{4}-\frac {3 \cos \left (x \right ) \sin \left (x \right ) b}{8}+\frac {\sin \left (x \right )^{4} a}{4}+\frac {3 b x}{8} \] Input:
int(sin(x)^3*(a*cos(x)+b*sin(x)),x)
Output:
( - 2*cos(x)*sin(x)**3*b - 3*cos(x)*sin(x)*b + 2*sin(x)**4*a + 3*b*x)/8