Integrand size = 27, antiderivative size = 73 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^4(c+d x)}{4 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^7(c+d x)}{7 d} \] Output:
1/4*a^3*sin(d*x+c)^4/d+3/5*a^3*sin(d*x+c)^5/d+1/2*a^3*sin(d*x+c)^6/d+1/7*a ^3*sin(d*x+c)^7/d
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 (-350+805 \cos (2 (c+d x))-280 \cos (4 (c+d x))+35 \cos (6 (c+d x))-1015 \sin (c+d x)+525 \sin (3 (c+d x))-119 \sin (5 (c+d x))+5 \sin (7 (c+d x)))}{2240 d} \] Input:
Integrate[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
-1/2240*(a^3*(-350 + 805*Cos[2*(c + d*x)] - 280*Cos[4*(c + d*x)] + 35*Cos[ 6*(c + d*x)] - 1015*Sin[c + d*x] + 525*Sin[3*(c + d*x)] - 119*Sin[5*(c + d *x)] + 5*Sin[7*(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.
\(\displaystyle \int \sin ^3(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^3 \cos (c+d x) (a \sin (c+d x)+a)^3dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \sin ^3(c+d x) (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int a^3 \sin ^3(c+d x) (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a^4 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\sin ^6(c+d x) a^6+3 \sin ^5(c+d x) a^6+3 \sin ^4(c+d x) a^6+\sin ^3(c+d x) a^6\right )d(a \sin (c+d x))}{a^4 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{7} a^7 \sin ^7(c+d x)+\frac {1}{2} a^7 \sin ^6(c+d x)+\frac {3}{5} a^7 \sin ^5(c+d x)+\frac {1}{4} a^7 \sin ^4(c+d x)}{a^4 d}\) |
Input:
Int[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
((a^7*Sin[c + d*x]^4)/4 + (3*a^7*Sin[c + d*x]^5)/5 + (a^7*Sin[c + d*x]^6)/ 2 + (a^7*Sin[c + d*x]^7)/7)/(a^4*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 = 224.49 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \sin \left (d x +c \right )^{7}}{7}+\frac {a^{3} \sin \left (d x +c \right )^{6}}{2}+\frac {3 a^{3} \sin \left (d x +c \right )^{5}}{5}+\frac {a^{3} \sin \left (d x +c \right )^{4}}{4}}{d}\) | \(58\) |
default | \(\frac {\frac {a^{3} \sin \left (d x +c \right )^{7}}{7}+\frac {a^{3} \sin \left (d x +c \right )^{6}}{2}+\frac {3 a^{3} \sin \left (d x +c \right )^{5}}{5}+\frac {a^{3} \sin \left (d x +c \right )^{4}}{4}}{d}\) | \(58\) |
parallelrisch | \(\frac {a^{3} \left (560+119 \sin \left (5 d x +5 c \right )-35 \cos \left (6 d x +6 c \right )-5 \sin \left (7 d x +7 c \right )-805 \cos \left (2 d x +2 c \right )+1015 \sin \left (d x +c \right )-525 \sin \left (3 d x +3 c \right )+280 \cos \left (4 d x +4 c \right )\right )}{2240 d}\) | \(85\) |
risch | \(\frac {29 a^{3} \sin \left (d x +c \right )}{64 d}-\frac {a^{3} \sin \left (7 d x +7 c \right )}{448 d}-\frac {a^{3} \cos \left (6 d x +6 c \right )}{64 d}+\frac {17 a^{3} \sin \left (5 d x +5 c \right )}{320 d}+\frac {a^{3} \cos \left (4 d x +4 c \right )}{8 d}-\frac {15 a^{3} \sin \left (3 d x +3 c \right )}{64 d}-\frac {23 a^{3} \cos \left (2 d x +2 c \right )}{64 d}\) | \(118\) |
norman | \(\frac {\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}+\frac {44 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}+\frac {44 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {1984 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{5 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(151\) |
orering | \(\text {Expression too large to display}\) | \(2286\) |
Input:
int(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/7*a^3*sin(d*x+c)^7+1/2*a^3*sin(d*x+c)^6+3/5*a^3*sin(d*x+c)^5+1/4*a^ 3*sin(d*x+c)^4)
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.34 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {70 \, a^{3} \cos \left (d x + c\right )^{6} - 245 \, a^{3} \cos \left (d x + c\right )^{4} + 280 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{6} - 36 \, a^{3} \cos \left (d x + c\right )^{4} + 57 \, a^{3} \cos \left (d x + c\right )^{2} - 26 \, a^{3}\right )} \sin \left (d x + c\right )}{140 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/140*(70*a^3*cos(d*x + c)^6 - 245*a^3*cos(d*x + c)^4 + 280*a^3*cos(d*x + c)^2 + 4*(5*a^3*cos(d*x + c)^6 - 36*a^3*cos(d*x + c)^4 + 57*a^3*cos(d*x + c)^2 - 26*a^3)*sin(d*x + c))/d
Time = 0.50 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac {a^{3} \sin ^{6}{\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)**3*(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((a**3*sin(c + d*x)**7/(7*d) + a**3*sin(c + d*x)**6/(2*d) + 3*a** 3*sin(c + d*x)**5/(5*d) + a**3*sin(c + d*x)**4/(4*d), Ne(d, 0)), (x*(a*sin (c) + a)**3*sin(c)**3*cos(c), True))
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {20 \, a^{3} \sin \left (d x + c\right )^{7} + 70 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} \sin \left (d x + c\right )^{4}}{140 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/140*(20*a^3*sin(d*x + c)^7 + 70*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c) ^5 + 35*a^3*sin(d*x + c)^4)/d
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {20 \, a^{3} \sin \left (d x + c\right )^{7} + 70 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} \sin \left (d x + c\right )^{4}}{140 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/140*(20*a^3*sin(d*x + c)^7 + 70*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c) ^5 + 35*a^3*sin(d*x + c)^4)/d
Time = 17.96 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a^3\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a^3\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \] Input:
int(cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^3,x)
Output:
((a^3*sin(c + d*x)^4)/4 + (3*a^3*sin(c + d*x)^5)/5 + (a^3*sin(c + d*x)^6)/ 2 + (a^3*sin(c + d*x)^7)/7)/d
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\sin \left (d x +c \right )^{4} a^{3} \left (20 \sin \left (d x +c \right )^{3}+70 \sin \left (d x +c \right )^{2}+84 \sin \left (d x +c \right )+35\right )}{140 d} \] Input:
int(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^3,x)
Output:
(sin(c + d*x)**4*a**3*(20*sin(c + d*x)**3 + 70*sin(c + d*x)**2 + 84*sin(c + d*x) + 35))/(140*d)