Integrand size = 27, antiderivative size = 73 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^4(c+d x)}{4 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{6 d} \] Output:
1/3*a^3*sin(d*x+c)^3/d+3/4*a^3*sin(d*x+c)^4/d+3/5*a^3*sin(d*x+c)^5/d+1/6*a ^3*sin(d*x+c)^6/d
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 (-45+870 \cos (2 (c+d x))-240 \cos (4 (c+d x))+10 \cos (6 (c+d x))-1200 \sin (c+d x)+520 \sin (3 (c+d x))-72 \sin (5 (c+d x)))}{1920 d} \] Input:
Integrate[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
-1/1920*(a^3*(-45 + 870*Cos[2*(c + d*x)] - 240*Cos[4*(c + d*x)] + 10*Cos[6 *(c + d*x)] - 1200*Sin[c + d*x] + 520*Sin[3*(c + d*x)] - 72*Sin[5*(c + d*x )]))/d
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
\(\displaystyle \int \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^2 \cos (c+d x) (a \sin (c+d x)+a)^3dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \sin ^2(c+d x) (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int a^2 \sin ^2(c+d x) (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\sin ^5(c+d x) a^5+3 \sin ^4(c+d x) a^5+3 \sin ^3(c+d x) a^5+\sin ^2(c+d x) a^5\right )d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{6} a^6 \sin ^6(c+d x)+\frac {3}{5} a^6 \sin ^5(c+d x)+\frac {3}{4} a^6 \sin ^4(c+d x)+\frac {1}{3} a^6 \sin ^3(c+d x)}{a^3 d}\) |
Input:
Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
((a^6*Sin[c + d*x]^3)/3 + (3*a^6*Sin[c + d*x]^4)/4 + (3*a^6*Sin[c + d*x]^5 )/5 + (a^6*Sin[c + d*x]^6)/6)/(a^3*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 = 53.73 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \sin \left (d x +c \right )^{6}}{6}+\frac {3 a^{3} \sin \left (d x +c \right )^{5}}{5}+\frac {3 a^{3} \sin \left (d x +c \right )^{4}}{4}+\frac {a^{3} \sin \left (d x +c \right )^{3}}{3}}{d}\) | \(58\) |
default | \(\frac {\frac {a^{3} \sin \left (d x +c \right )^{6}}{6}+\frac {3 a^{3} \sin \left (d x +c \right )^{5}}{5}+\frac {3 a^{3} \sin \left (d x +c \right )^{4}}{4}+\frac {a^{3} \sin \left (d x +c \right )^{3}}{3}}{d}\) | \(58\) |
parallelrisch | \(\frac {a^{3} \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-76+36 \cos \left (2 d x +2 c \right )+5 \sin \left (3 d x +3 c \right )-105 \sin \left (d x +c \right )\right ) \left (\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}\) | \(83\) |
risch | \(\frac {5 a^{3} \sin \left (d x +c \right )}{8 d}-\frac {a^{3} \cos \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{3} \cos \left (4 d x +4 c \right )}{8 d}-\frac {13 a^{3} \sin \left (3 d x +3 c \right )}{48 d}-\frac {29 a^{3} \cos \left (2 d x +2 c \right )}{64 d}\) | \(101\) |
norman | \(\frac {\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {136 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {136 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 d}+\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {12 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {104 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(151\) |
orering | \(\text {Expression too large to display}\) | \(1594\) |
Input:
int(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/6*a^3*sin(d*x+c)^6+3/5*a^3*sin(d*x+c)^5+3/4*a^3*sin(d*x+c)^4+1/3*a^ 3*sin(d*x+c)^3)
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {10 \, a^{3} \cos \left (d x + c\right )^{6} - 75 \, a^{3} \cos \left (d x + c\right )^{4} + 120 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, {\left (9 \, a^{3} \cos \left (d x + c\right )^{4} - 23 \, a^{3} \cos \left (d x + c\right )^{2} + 14 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/60*(10*a^3*cos(d*x + c)^6 - 75*a^3*cos(d*x + c)^4 + 120*a^3*cos(d*x + c )^2 - 4*(9*a^3*cos(d*x + c)^4 - 23*a^3*cos(d*x + c)^2 + 14*a^3)*sin(d*x + c))/d
Time = 0.34 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} \sin ^{6}{\left (c + d x \right )}}{6 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{2}{\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((a**3*sin(c + d*x)**6/(6*d) + 3*a**3*sin(c + d*x)**5/(5*d) + 3*a **3*sin(c + d*x)**4/(4*d) + a**3*sin(c + d*x)**3/(3*d), Ne(d, 0)), (x*(a*s in(c) + a)**3*sin(c)**2*cos(c), True))
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3}}{60 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/60*(10*a^3*sin(d*x + c)^6 + 36*a^3*sin(d*x + c)^5 + 45*a^3*sin(d*x + c)^ 4 + 20*a^3*sin(d*x + c)^3)/d
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3}}{60 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/60*(10*a^3*sin(d*x + c)^6 + 36*a^3*sin(d*x + c)^5 + 45*a^3*sin(d*x + c)^ 4 + 20*a^3*sin(d*x + c)^3)/d
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a^3\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \] Input:
int(cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^3,x)
Output:
((a^3*sin(c + d*x)^3)/3 + (3*a^3*sin(c + d*x)^4)/4 + (3*a^3*sin(c + d*x)^5 )/5 + (a^3*sin(c + d*x)^6)/6)/d
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63 \[ \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\sin \left (d x +c \right )^{3} a^{3} \left (10 \sin \left (d x +c \right )^{3}+36 \sin \left (d x +c \right )^{2}+45 \sin \left (d x +c \right )+20\right )}{60 d} \] Input:
int(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
(sin(c + d*x)**3*a**3*(10*sin(c + d*x)**3 + 36*sin(c + d*x)**2 + 45*sin(c + d*x) + 20))/(60*d)