Integrand size = 25, antiderivative size = 55 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^4(c+d x)}{4 d} \] Output:
1/2*a^2*sin(d*x+c)^2/d+2/3*a^2*sin(d*x+c)^3/d+1/4*a^2*sin(d*x+c)^4/d
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1}{2} \left (\frac {a^2 \sin ^2(c+d x)}{d}+\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}\right ) \] Input:
Integrate[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]
Output:
((a^2*Sin[c + d*x]^2)/d + (4*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x] ^4)/(2*d))/2
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3312, 27, 49, 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.
\(\displaystyle \int \sin (c+d x) \cos (c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x) \cos (c+d x) (a \sin (c+d x)+a)^2dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \sin (c+d x) (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int a \sin (c+d x) (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^2 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\sin ^3(c+d x) a^3+2 \sin ^2(c+d x) a^3+\sin (c+d x) a^3\right )d(a \sin (c+d x))}{a^2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{4} a^4 \sin ^4(c+d x)+\frac {2}{3} a^4 \sin ^3(c+d x)+\frac {1}{2} a^4 \sin ^2(c+d x)}{a^2 d}\) |
Input:
Int[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]
Output:
((a^4*Sin[c + d*x]^2)/2 + (2*a^4*Sin[c + d*x]^3)/3 + (a^4*Sin[c + d*x]^4)/ 4)/(a^2*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 5.52 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \sin \left (d x +c \right )^{4}}{4}+\frac {2 a^{2} \sin \left (d x +c \right )^{3}}{3}+\frac {a^{2} \sin \left (d x +c \right )^{2}}{2}}{d}\) | \(45\) |
default | \(\frac {\frac {a^{2} \sin \left (d x +c \right )^{4}}{4}+\frac {2 a^{2} \sin \left (d x +c \right )^{3}}{3}+\frac {a^{2} \sin \left (d x +c \right )^{2}}{2}}{d}\) | \(45\) |
parallelrisch | \(\frac {a^{2} \left (-16 \sin \left (3 d x +3 c \right )+48 \sin \left (d x +c \right )+3 \cos \left (4 d x +4 c \right )+33-36 \cos \left (2 d x +2 c \right )\right )}{96 d}\) | \(52\) |
risch | \(\frac {a^{2} \sin \left (d x +c \right )}{2 d}+\frac {a^{2} \cos \left (4 d x +4 c \right )}{32 d}-\frac {a^{2} \sin \left (3 d x +3 c \right )}{6 d}-\frac {3 a^{2} \cos \left (2 d x +2 c \right )}{8 d}\) | \(67\) |
norman | \(\frac {\frac {16 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {16 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}+\frac {8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(113\) |
orering | \(-\frac {205 \left (-d \sin \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}+\cos \left (d x +c \right )^{2} d \left (a +a \sin \left (d x +c \right )\right )^{2}+2 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right ) a d \right )}{144 d^{2}}-\frac {91 \left (-4 d^{3} \cos \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}-32 d^{3} \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{2} a +4 d^{3} \sin \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}-12 d^{3} \sin \left (d x +c \right )^{2} a^{2} \cos \left (d x +c \right )^{2}+6 d^{3} \sin \left (d x +c \right )^{3} \left (a +a \sin \left (d x +c \right )\right ) a +6 \cos \left (d x +c \right )^{4} d^{3} a^{2}\right )}{192 d^{4}}-\frac {5 \left (-16 d^{5} \sin \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}+332 d^{5} \cos \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right ) \sin \left (d x +c \right ) a +16 d^{5} \cos \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}-120 \cos \left (d x +c \right )^{4} a^{2} d^{5}+420 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2} a^{2} d^{5}-90 d^{5} \sin \left (d x +c \right )^{3} \left (a +a \sin \left (d x +c \right )\right ) a -30 \sin \left (d x +c \right )^{4} a^{2} d^{5}\right )}{96 d^{6}}-\frac {-64 d^{7} \cos \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}-3152 d^{7} \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{2} a +64 d^{7} \sin \left (d x +c \right )^{2} \left (a +a \sin \left (d x +c \right )\right )^{2}-9072 d^{7} \sin \left (d x +c \right )^{2} a^{2} \cos \left (d x +c \right )^{2}+966 d^{7} \sin \left (d x +c \right )^{3} \left (a +a \sin \left (d x +c \right )\right ) a +2016 d^{7} \cos \left (d x +c \right )^{4} a^{2}+1050 \sin \left (d x +c \right )^{4} d^{7} a^{2}}{576 d^{8}}\) | \(563\) |
Input:
int(cos(d*x+c)*sin(d*x+c)*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/4*a^2*sin(d*x+c)^4+2/3*a^2*sin(d*x+c)^3+1/2*a^2*sin(d*x+c)^2)
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \cos \left (d x + c\right )^{4} - 12 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{12 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Output:
1/12*(3*a^2*cos(d*x + c)^4 - 12*a^2*cos(d*x + c)^2 - 8*(a^2*cos(d*x + c)^2 - a^2)*sin(d*x + c))/d
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin {\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)*(a+a*sin(d*x+c))**2,x)
Output:
Piecewise((a**2*sin(c + d*x)**4/(4*d) + 2*a**2*sin(c + d*x)**3/(3*d) - a** 2*cos(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)*cos(c), Tr ue))
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \sin \left (d x + c\right )^{4} + 8 \, a^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2}}{12 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="maxima")
Output:
1/12*(3*a^2*sin(d*x + c)^4 + 8*a^2*sin(d*x + c)^3 + 6*a^2*sin(d*x + c)^2)/ d
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \sin \left (d x + c\right )^{4} + 8 \, a^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2}}{12 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/12*(3*a^2*sin(d*x + c)^4 + 8*a^2*sin(d*x + c)^3 + 6*a^2*sin(d*x + c)^2)/ d
Time = 17.97 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\sin \left (c+d\,x\right )}^2\,\left (3\,{\sin \left (c+d\,x\right )}^2+8\,\sin \left (c+d\,x\right )+6\right )}{12\,d} \] Input:
int(cos(c + d*x)*sin(c + d*x)*(a + a*sin(c + d*x))^2,x)
Output:
(a^2*sin(c + d*x)^2*(8*sin(c + d*x) + 3*sin(c + d*x)^2 + 6))/(12*d)
Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-6 \cos \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )^{4}+8 \sin \left (d x +c \right )^{3}\right )}{12 d} \] Input:
int(cos(d*x+c)*sin(d*x+c)*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*( - 6*cos(c + d*x)**2 + 3*sin(c + d*x)**4 + 8*sin(c + d*x)**3))/(12* d)