Integrand size = 27, antiderivative size = 65 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^7(c+d x)}{7 d} \] Output:
1/4*a*sin(d*x+c)^4/d+1/5*a*sin(d*x+c)^5/d-1/6*a*sin(d*x+c)^6/d-1/7*a*sin(d *x+c)^7/d
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-315 \cos (2 (c+d x))+35 \cos (6 (c+d x))+96 (9+5 \cos (2 (c+d x))) \sin ^5(c+d x)\right )}{6720 d} \] Input:
Integrate[Cos[c + d*x]^3*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
(a*(-315*Cos[2*(c + d*x)] + 35*Cos[6*(c + d*x)] + 96*(9 + 5*Cos[2*(c + d*x )])*Sin[c + d*x]^5))/(6720*d)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 84, 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.
\(\displaystyle \int \sin ^3(c+d x) \cos ^3(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^3 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \sin ^3(c+d x) (a-a \sin (c+d x)) (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int a^3 \sin ^3(c+d x) (a-a \sin (c+d x)) (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^6 d}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle \frac {\int \left (-\sin ^6(c+d x) a^6-\sin ^5(c+d x) a^6+\sin ^4(c+d x) a^6+\sin ^3(c+d x) a^6\right )d(a \sin (c+d x))}{a^6 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{7} a^7 \sin ^7(c+d x)-\frac {1}{6} a^7 \sin ^6(c+d x)+\frac {1}{5} a^7 \sin ^5(c+d x)+\frac {1}{4} a^7 \sin ^4(c+d x)}{a^6 d}\) |
Input:
Int[Cos[c + d*x]^3*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
((a^7*Sin[c + d*x]^4)/4 + (a^7*Sin[c + d*x]^5)/5 - (a^7*Sin[c + d*x]^6)/6 - (a^7*Sin[c + d*x]^7)/7)/(a^6*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 15.63 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(48\) |
default | \(-\frac {a \left (\frac {\sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{4}}{4}\right )}{d}\) | \(48\) |
parallelrisch | \(\frac {a \left (280-21 \sin \left (5 d x +5 c \right )+35 \cos \left (6 d x +6 c \right )+15 \sin \left (7 d x +7 c \right )-315 \cos \left (2 d x +2 c \right )+315 \sin \left (d x +c \right )-105 \sin \left (3 d x +3 c \right )\right )}{6720 d}\) | \(72\) |
risch | \(\frac {3 a \sin \left (d x +c \right )}{64 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \cos \left (6 d x +6 c \right )}{192 d}-\frac {a \sin \left (5 d x +5 c \right )}{320 d}-\frac {a \sin \left (3 d x +3 c \right )}{64 d}-\frac {3 a \cos \left (2 d x +2 c \right )}{64 d}\) | \(89\) |
norman | \(\frac {\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}+\frac {32 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}-\frac {192 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {32 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{5 d}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(137\) |
orering | \(\text {Expression too large to display}\) | \(1031\) |
Input:
int(cos(d*x+c)^3*sin(d*x+c)^3*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/7*sin(d*x+c)^7+1/6*sin(d*x+c)^6-1/5*sin(d*x+c)^5-1/4*sin(d*x+c)^4)
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {70 \, a \cos \left (d x + c\right )^{6} - 105 \, a \cos \left (d x + c\right )^{4} + 12 \, {\left (5 \, a \cos \left (d x + c\right )^{6} - 8 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right )}{420 \, d} \] Input:
integrate(cos(d*x+c)^3*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/420*(70*a*cos(d*x + c)^6 - 105*a*cos(d*x + c)^4 + 12*(5*a*cos(d*x + c)^6 - 8*a*cos(d*x + c)^4 + a*cos(d*x + c)^2 + 2*a)*sin(d*x + c))/d
Time = 0.47 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {2 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{6}{\left (c + d x \right )}}{12 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**3*sin(d*x+c)**3*(a+a*sin(d*x+c)),x)
Output:
Piecewise((2*a*sin(c + d*x)**7/(35*d) + a*sin(c + d*x)**6/(12*d) + a*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + a*sin(c + d*x)**4*cos(c + d*x)**2/(4*d) , Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**3*cos(c)**3, True))
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 84 \, a \sin \left (d x + c\right )^{5} - 105 \, a \sin \left (d x + c\right )^{4}}{420 \, d} \] Input:
integrate(cos(d*x+c)^3*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/420*(60*a*sin(d*x + c)^7 + 70*a*sin(d*x + c)^6 - 84*a*sin(d*x + c)^5 - 105*a*sin(d*x + c)^4)/d
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 84 \, a \sin \left (d x + c\right )^{5} - 105 \, a \sin \left (d x + c\right )^{4}}{420 \, d} \] Input:
integrate(cos(d*x+c)^3*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/420*(60*a*sin(d*x + c)^7 + 70*a*sin(d*x + c)^6 - 84*a*sin(d*x + c)^5 - 105*a*sin(d*x + c)^4)/d
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \] Input:
int(cos(c + d*x)^3*sin(c + d*x)^3*(a + a*sin(c + d*x)),x)
Output:
((a*sin(c + d*x)^4)/4 + (a*sin(c + d*x)^5)/5 - (a*sin(c + d*x)^6)/6 - (a*s in(c + d*x)^7)/7)/d
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68 \[ \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\sin \left (d x +c \right )^{4} a \left (-60 \sin \left (d x +c \right )^{3}-70 \sin \left (d x +c \right )^{2}+84 \sin \left (d x +c \right )+105\right )}{420 d} \] Input:
int(cos(d*x+c)^3*sin(d*x+c)^3*(a+a*sin(d*x+c)),x)
Output:
(sin(c + d*x)**4*a*( - 60*sin(c + d*x)**3 - 70*sin(c + d*x)**2 + 84*sin(c + d*x) + 105))/(420*d)