Integrand size = 27, antiderivative size = 105 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a x}{16}-\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^3(c+d x) \sin ^3(c+d x)}{6 d} \] Output:
1/16*a*x-1/3*a*cos(d*x+c)^3/d+1/5*a*cos(d*x+c)^5/d+1/16*a*cos(d*x+c)*sin(d *x+c)/d-1/8*a*cos(d*x+c)^3*sin(d*x+c)/d-1/6*a*cos(d*x+c)^3*sin(d*x+c)^3/d
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.68 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (60 d x-120 \cos (c+d x)-20 \cos (3 (c+d x))+12 \cos (5 (c+d x))-15 \sin (2 (c+d x))-15 \sin (4 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \] Input:
Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
(a*(60*d*x - 120*Cos[c + d*x] - 20*Cos[3*(c + d*x)] + 12*Cos[5*(c + d*x)] - 15*Sin[2*(c + d*x)] - 15*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)
Time = 0.60 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3317, 3042, 3045, 244, 2009, 3048, 3042, 3048, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^2 (a \sin (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^2(c+d x) \sin ^4(c+d x)dx+a \int \cos ^2(c+d x) \sin ^3(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \cos (c+d x)^2 \sin (c+d x)^3dx+a \int \cos (c+d x)^2 \sin (c+d x)^4dx\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle a \int \cos (c+d x)^2 \sin (c+d x)^4dx-\frac {a \int \cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle a \int \cos (c+d x)^2 \sin (c+d x)^4dx-\frac {a \int \left (\cos ^2(c+d x)-\cos ^4(c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \cos (c+d x)^2 \sin (c+d x)^4dx-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {1}{2} \int \cos ^2(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {1}{2} \int \cos (c+d x)^2 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {1}{2} \left (\frac {1}{4} \int \cos ^2(c+d x)dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {1}{2} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 d}\right )-\frac {a \left (\frac {1}{3} \cos ^3(c+d x)-\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]
Output:
-((a*(Cos[c + d*x]^3/3 - Cos[c + d*x]^5/5))/d) + a*(-1/6*(Cos[c + d*x]^3*S in[c + d*x]^3)/d + (-1/4*(Cos[c + d*x]^3*Sin[c + d*x])/d + (x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d))/4)/2)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 6.82 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {a \left (60 d x +12 \cos \left (5 d x +5 c \right )-20 \cos \left (3 d x +3 c \right )-15 \sin \left (4 d x +4 c \right )-15 \sin \left (2 d x +2 c \right )-120 \cos \left (d x +c \right )+5 \sin \left (6 d x +6 c \right )-128\right )}{960 d}\) | \(76\) |
| risch | \(\frac {a x}{16}-\frac {a \cos \left (d x +c \right )}{8 d}+\frac {a \sin \left (6 d x +6 c \right )}{192 d}+\frac {a \cos \left (5 d x +5 c \right )}{80 d}-\frac {a \sin \left (4 d x +4 c \right )}{64 d}-\frac {a \cos \left (3 d x +3 c \right )}{48 d}-\frac {a \sin \left (2 d x +2 c \right )}{64 d}\) | \(93\) |
| derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )}{d}\) | \(95\) |
| default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )}{d}\) | \(95\) |
| norman | \(\frac {\frac {a x}{16}-\frac {4 a}{15 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {17 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {19 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {19 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {17 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {15 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {5 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {15 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}-\frac {8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(269\) |
| orering | \(\text {Expression too large to display}\) | \(1810\) |
Input:
int(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/960*a*(60*d*x+12*cos(5*d*x+5*c)-20*cos(3*d*x+3*c)-15*sin(4*d*x+4*c)-15*s in(2*d*x+2*c)-120*cos(d*x+c)+5*sin(6*d*x+6*c)-128)/d
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {48 \, a \cos \left (d x + c\right )^{5} - 80 \, a \cos \left (d x + c\right )^{3} + 15 \, a d x + 5 \, {\left (8 \, a \cos \left (d x + c\right )^{5} - 14 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/240*(48*a*cos(d*x + c)^5 - 80*a*cos(d*x + c)^3 + 15*a*d*x + 5*(8*a*cos(d *x + c)^5 - 14*a*cos(d*x + c)^3 + 3*a*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (92) = 184\).
Time = 0.36 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.83 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*sin(d*x+c)**3*(a+a*sin(d*x+c)),x)
Output:
Piecewise((a*x*sin(c + d*x)**6/16 + 3*a*x*sin(c + d*x)**4*cos(c + d*x)**2/ 16 + 3*a*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + a*x*cos(c + d*x)**6/16 + a *sin(c + d*x)**5*cos(c + d*x)/(16*d) - a*sin(c + d*x)**3*cos(c + d*x)**3/( 6*d) - a*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - a*sin(c + d*x)*cos(c + d* x)**5/(16*d) - 2*a*cos(c + d*x)**5/(15*d), Ne(d, 0)), (x*(a*sin(c) + a)*si n(c)**3*cos(c)**2, True))
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.62 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {64 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{960 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/960*(64*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a - 5*(4*sin(2*d*x + 2*c)^ 3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a)/d
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1}{16} \, a x + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {a \cos \left (d x + c\right )}{8 \, d} + \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/16*a*x + 1/80*a*cos(5*d*x + 5*c)/d - 1/48*a*cos(3*d*x + 3*c)/d - 1/8*a*c os(d*x + c)/d + 1/192*a*sin(6*d*x + 6*c)/d - 1/64*a*sin(4*d*x + 4*c)/d - 1 /64*a*sin(2*d*x + 2*c)/d
Time = 20.77 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.15 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,x}{16}+\frac {\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\left (\frac {a\,\left (225\,c+225\,d\,x-960\right )}{240}-\frac {15\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {19\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\left (\frac {a\,\left (300\,c+300\,d\,x-640\right )}{240}-\frac {5\,a\,\left (c+d\,x\right )}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {19\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\left (\frac {a\,\left (90\,c+90\,d\,x-384\right )}{240}-\frac {3\,a\,\left (c+d\,x\right )}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {a\,\left (15\,c+15\,d\,x-64\right )}{240}-\frac {a\,\left (c+d\,x\right )}{16}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int(cos(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x)),x)
Output:
(a*x)/16 + ((a*(15*c + 15*d*x - 64))/240 - (a*tan(c/2 + (d*x)/2))/8 - (a*( c + d*x))/16 + tan(c/2 + (d*x)/2)^2*((a*(90*c + 90*d*x - 384))/240 - (3*a* (c + d*x))/8) + tan(c/2 + (d*x)/2)^6*((a*(300*c + 300*d*x - 640))/240 - (5 *a*(c + d*x))/4) + tan(c/2 + (d*x)/2)^8*((a*(225*c + 225*d*x - 960))/240 - (15*a*(c + d*x))/16) - (17*a*tan(c/2 + (d*x)/2)^3)/24 + (19*a*tan(c/2 + ( d*x)/2)^5)/4 - (19*a*tan(c/2 + (d*x)/2)^7)/4 + (17*a*tan(c/2 + (d*x)/2)^9) /24 + (a*tan(c/2 + (d*x)/2)^11)/8)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^6)
Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )-32 \cos \left (d x +c \right )+15 d x +32\right )}{240 d} \] Input:
int(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c)),x)
Output:
(a*(40*cos(c + d*x)*sin(c + d*x)**5 + 48*cos(c + d*x)*sin(c + d*x)**4 - 10 *cos(c + d*x)*sin(c + d*x)**3 - 16*cos(c + d*x)*sin(c + d*x)**2 - 15*cos(c + d*x)*sin(c + d*x) - 32*cos(c + d*x) + 15*d*x + 32))/(240*d)