Integrand size = 29, antiderivative size = 135 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 x}{8}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^2 \cos ^3(c+d x) \sin ^3(c+d x)}{3 d} \] Output:
1/8*a^2*x-2/3*a^2*cos(d*x+c)^3/d+3/5*a^2*cos(d*x+c)^5/d-1/7*a^2*cos(d*x+c) ^7/d+1/8*a^2*cos(d*x+c)*sin(d*x+c)/d-1/4*a^2*cos(d*x+c)^3*sin(d*x+c)/d-1/3 *a^2*cos(d*x+c)^3*sin(d*x+c)^3/d
Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (840 c+840 d x-1365 \cos (c+d x)-175 \cos (3 (c+d x))+147 \cos (5 (c+d x))-15 \cos (7 (c+d x))-210 \sin (2 (c+d x))-210 \sin (4 (c+d x))+70 \sin (6 (c+d x)))}{6720 d} \] Input:
Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*(840*c + 840*d*x - 1365*Cos[c + d*x] - 175*Cos[3*(c + d*x)] + 147*Cos [5*(c + d*x)] - 15*Cos[7*(c + d*x)] - 210*Sin[2*(c + d*x)] - 210*Sin[4*(c + d*x)] + 70*Sin[6*(c + d*x)]))/(6720*d)
Time = 0.49 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^2(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^2 (a \sin (c+d x)+a)^2dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \sin ^5(c+d x) \cos ^2(c+d x)+2 a^2 \sin ^4(c+d x) \cos ^2(c+d x)+a^2 \sin ^3(c+d x) \cos ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \sin ^3(c+d x) \cos ^3(c+d x)}{3 d}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^2 x}{8}\) |
Input:
Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*x)/8 - (2*a^2*Cos[c + d*x]^3)/(3*d) + (3*a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cos[c + d*x]^7)/(7*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^2* Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x]^3)/( 3*d)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 82.58 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66
| method | result | size |
| parallelrisch | \(\frac {a^{2} \left (840 d x +70 \sin \left (6 d x +6 c \right )-210 \sin \left (4 d x +4 c \right )-210 \sin \left (2 d x +2 c \right )+147 \cos \left (5 d x +5 c \right )-175 \cos \left (3 d x +3 c \right )-1365 \cos \left (d x +c \right )-15 \cos \left (7 d x +7 c \right )-1408\right )}{6720 d}\) | \(89\) |
| risch | \(\frac {a^{2} x}{8}-\frac {13 a^{2} \cos \left (d x +c \right )}{64 d}-\frac {a^{2} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{96 d}+\frac {7 a^{2} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{2} \sin \left (4 d x +4 c \right )}{32 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right )}{192 d}-\frac {a^{2} \sin \left (2 d x +2 c \right )}{32 d}\) | \(124\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{3}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a^{2} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )}{d}\) | \(151\) |
| default | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{3}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a^{2} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )}{d}\) | \(151\) |
| norman | \(\frac {\frac {a^{2} x}{8}-\frac {44 a^{2}}{105 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {5 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {97 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {97 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}+\frac {5 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}+\frac {7 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {21 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8}+\frac {35 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8}+\frac {35 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {21 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {7 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8}+\frac {a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}+\frac {8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {44 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 d}-\frac {52 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(358\) |
| orering | \(\text {Expression too large to display}\) | \(3660\) |
Input:
int(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/6720*a^2*(840*d*x+70*sin(6*d*x+6*c)-210*sin(4*d*x+4*c)-210*sin(2*d*x+2*c )+147*cos(5*d*x+5*c)-175*cos(3*d*x+3*c)-1365*cos(d*x+c)-15*cos(7*d*x+7*c)- 1408)/d
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.73 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {120 \, a^{2} \cos \left (d x + c\right )^{7} - 504 \, a^{2} \cos \left (d x + c\right )^{5} + 560 \, a^{2} \cos \left (d x + c\right )^{3} - 105 \, a^{2} d x - 35 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{5} - 14 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/840*(120*a^2*cos(d*x + c)^7 - 504*a^2*cos(d*x + c)^5 + 560*a^2*cos(d*x + c)^3 - 105*a^2*d*x - 35*(8*a^2*cos(d*x + c)^5 - 14*a^2*cos(d*x + c)^3 + 3*a^2*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (121) = 242\).
Time = 0.53 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.04 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{2} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {8 a^{2} \cos ^{7}{\left (c + d x \right )}}{105 d} - \frac {2 a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*sin(d*x+c)**3*(a+a*sin(d*x+c))**2,x)
Output:
Piecewise((a**2*x*sin(c + d*x)**6/8 + 3*a**2*x*sin(c + d*x)**4*cos(c + d*x )**2/8 + 3*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/8 + a**2*x*cos(c + d*x)* *6/8 + a**2*sin(c + d*x)**5*cos(c + d*x)/(8*d) - a**2*sin(c + d*x)**4*cos( c + d*x)**3/(3*d) - a**2*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) - 4*a**2*si n(c + d*x)**2*cos(c + d*x)**5/(15*d) - a**2*sin(c + d*x)**2*cos(c + d*x)** 3/(3*d) - a**2*sin(c + d*x)*cos(c + d*x)**5/(8*d) - 8*a**2*cos(c + d*x)**7 /(105*d) - 2*a**2*cos(c + d*x)**5/(15*d), Ne(d, 0)), (x*(a*sin(c) + a)**2* sin(c)**3*cos(c)**2, True))
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {32 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{2} - 224 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{3360 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/3360*(32*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^ 2 - 224*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^2 + 35*(4*sin(2*d*x + 2*c) ^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^2)/d
Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.91 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1}{8} \, a^{2} x - \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, a^{2} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {13 \, a^{2} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/8*a^2*x - 1/448*a^2*cos(7*d*x + 7*c)/d + 7/320*a^2*cos(5*d*x + 5*c)/d - 5/192*a^2*cos(3*d*x + 3*c)/d - 13/64*a^2*cos(d*x + c)/d + 1/96*a^2*sin(6*d *x + 6*c)/d - 1/32*a^2*sin(4*d*x + 4*c)/d - 1/32*a^2*sin(2*d*x + 2*c)/d
Time = 20.89 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.45 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,x}{8}-\frac {\frac {a^2\,\left (c+d\,x\right )}{8}+\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {97\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {97\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}-\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}-\frac {a^2\,\left (105\,c+105\,d\,x-352\right )}{840}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {7\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (735\,c+735\,d\,x-2464\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (2205\,c+2205\,d\,x-3360\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (2205\,c+2205\,d\,x-4032\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (3675\,c+3675\,d\,x+2240\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (3675\,c+3675\,d\,x-14560\right )}{840}\right )+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \] Input:
int(cos(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^2,x)
Output:
(a^2*x)/8 - ((a^2*(c + d*x))/8 + (5*a^2*tan(c/2 + (d*x)/2)^3)/3 - (97*a^2* tan(c/2 + (d*x)/2)^5)/12 + (97*a^2*tan(c/2 + (d*x)/2)^9)/12 - (5*a^2*tan(c /2 + (d*x)/2)^11)/3 - (a^2*tan(c/2 + (d*x)/2)^13)/4 - (a^2*(105*c + 105*d* x - 352))/840 + tan(c/2 + (d*x)/2)^2*((7*a^2*(c + d*x))/8 - (a^2*(735*c + 735*d*x - 2464))/840) + tan(c/2 + (d*x)/2)^10*((21*a^2*(c + d*x))/8 - (a^2 *(2205*c + 2205*d*x - 3360))/840) + tan(c/2 + (d*x)/2)^4*((21*a^2*(c + d*x ))/8 - (a^2*(2205*c + 2205*d*x - 4032))/840) + tan(c/2 + (d*x)/2)^6*((35*a ^2*(c + d*x))/8 - (a^2*(3675*c + 3675*d*x + 2240))/840) + tan(c/2 + (d*x)/ 2)^8*((35*a^2*(c + d*x))/8 - (a^2*(3675*c + 3675*d*x - 14560))/840) + (a^2 *tan(c/2 + (d*x)/2))/4)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^7)
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-70 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-88 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )-176 \cos \left (d x +c \right )+105 d x +176\right )}{840 d} \] Input:
int(cos(d*x+c)^2*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(120*cos(c + d*x)*sin(c + d*x)**6 + 280*cos(c + d*x)*sin(c + d*x)**5 + 144*cos(c + d*x)*sin(c + d*x)**4 - 70*cos(c + d*x)*sin(c + d*x)**3 - 88 *cos(c + d*x)*sin(c + d*x)**2 - 105*cos(c + d*x)*sin(c + d*x) - 176*cos(c + d*x) + 105*d*x + 176))/(840*d)