Integrand size = 27, antiderivative size = 83 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\log (1+\sin (c+d x))}{a^4 d}+\frac {1}{3 a d (a+a \sin (c+d x))^3}-\frac {3}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {3}{d \left (a^4+a^4 \sin (c+d x)\right )} \] Output:
ln(1+sin(d*x+c))/a^4/d+1/3/a/d/(a+a*sin(d*x+c))^3-3/2/d/(a^2+a^2*sin(d*x+c ))^2+3/d/(a^4+a^4*sin(d*x+c))
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {11+27 \sin (c+d x)+18 \sin ^2(c+d x)+6 \log (1+\sin (c+d x)) (1+\sin (c+d x))^3}{6 a^4 d (1+\sin (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]
Output:
(11 + 27*Sin[c + d*x] + 18*Sin[c + d*x]^2 + 6*Log[1 + Sin[c + d*x]]*(1 + S in[c + d*x])^3)/(6*a^4*d*(1 + Sin[c + d*x])^3)
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos (c+d x)}{(a \sin (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)}{(a \sin (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {\sin ^3(c+d x)}{(\sin (c+d x) a+a)^4}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^3 \sin ^3(c+d x)}{(\sin (c+d x) a+a)^4}d(a \sin (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\int \left (-\frac {a^3}{(\sin (c+d x) a+a)^4}+\frac {3 a^2}{(\sin (c+d x) a+a)^3}-\frac {3 a}{(\sin (c+d x) a+a)^2}+\frac {1}{\sin (c+d x) a+a}\right )d(a \sin (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^3}{3 (a \sin (c+d x)+a)^3}-\frac {3 a^2}{2 (a \sin (c+d x)+a)^2}+\frac {3 a}{a \sin (c+d x)+a}+\log (a \sin (c+d x)+a)}{a^4 d}\) |
Input:
Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]
Output:
(Log[a + a*Sin[c + d*x]] + a^3/(3*(a + a*Sin[c + d*x])^3) - (3*a^2)/(2*(a + a*Sin[c + d*x])^2) + (3*a)/(a + a*Sin[c + d*x]))/(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 = 0.67 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {3}{1+\sin \left (d x +c \right )}+\ln \left (1+\sin \left (d x +c \right )\right )-\frac {3}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{4}}\) | \(54\) |
| default | \(\frac {\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {3}{1+\sin \left (d x +c \right )}+\ln \left (1+\sin \left (d x +c \right )\right )-\frac {3}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{4}}\) | \(54\) |
| risch | \(-\frac {i x}{a^{4}}-\frac {2 i c}{d \,a^{4}}+\frac {2 i \left (-40 \,{\mathrm e}^{3 i \left (d x +c \right )}-27 i {\mathrm e}^{2 i \left (d x +c \right )}+27 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{6}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{4}}\) | \(121\) |
| parallelrisch | \(\frac {\left (-36 \cos \left (2 d x +2 c \right )+90 \sin \left (d x +c \right )-6 \sin \left (3 d x +3 c \right )+60\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (72 \cos \left (2 d x +2 c \right )-180 \sin \left (d x +c \right )+12 \sin \left (3 d x +3 c \right )-120\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-30 \cos \left (2 d x +2 c \right )+57 \sin \left (d x +c \right )-11 \sin \left (3 d x +3 c \right )+30}{6 d \,a^{4} \left (\sin \left (3 d x +3 c \right )-15 \sin \left (d x +c \right )-10+6 \cos \left (2 d x +2 c \right )\right )}\) | \(163\) |
| norman | \(\frac {-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d a}-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d a}-\frac {228 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d a}-\frac {228 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}-\frac {230 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}-\frac {230 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d a}-\frac {110 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d a}-\frac {110 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d a}-\frac {416 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d a}-\frac {416 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d a}-\frac {566 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d a}-\frac {566 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}-\frac {\ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \,a^{4}}\) | \(341\) |
Input:
int(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d/a^4*(1/3/(1+sin(d*x+c))^3+3/(1+sin(d*x+c))+ln(1+sin(d*x+c))-3/2/(1+sin (d*x+c))^2)
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {18 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 27 \, \sin \left (d x + c\right ) - 29}{6 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="fricas" )
Output:
1/6*(18*cos(d*x + c)^2 + 6*(3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 4)*sin(d* x + c) - 4)*log(sin(d*x + c) + 1) - 27*sin(d*x + c) - 29)/(3*a^4*d*cos(d*x + c)^2 - 4*a^4*d + (a^4*d*cos(d*x + c)^2 - 4*a^4*d)*sin(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (70) = 140\).
Time = 0.89 (sec) , antiderivative size = 466, normalized size of antiderivative = 5.61 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\begin {cases} \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {27 \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {11}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)**3/(a+a*sin(d*x+c))**4,x)
Output:
Piecewise((6*log(sin(c + d*x) + 1)*sin(c + d*x)**3/(6*a**4*d*sin(c + d*x)* *3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 18*l og(sin(c + d*x) + 1)*sin(c + d*x)**2/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d *sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 18*log(sin(c + d*x ) + 1)*sin(c + d*x)/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 6*log(sin(c + d*x) + 1)/(6*a**4*d*s in(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a* *4*d) + 18*sin(c + d*x)**2/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d *x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 27*sin(c + d*x)/(6*a**4*d*si n(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a** 4*d) + 11/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4* d*sin(c + d*x) + 6*a**4*d), Ne(d, 0)), (x*sin(c)**3*cos(c)/(a*sin(c) + a)* *4, True))
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="maxima" )
Output:
1/6*((18*sin(d*x + c)^2 + 27*sin(d*x + c) + 11)/(a^4*sin(d*x + c)^3 + 3*a^ 4*sin(d*x + c)^2 + 3*a^4*sin(d*x + c) + a^4) + 6*log(sin(d*x + c) + 1)/a^4 )/d
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} d} + \frac {18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{6 \, a^{4} d {\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
log(abs(sin(d*x + c) + 1))/(a^4*d) + 1/6*(18*sin(d*x + c)^2 + 27*sin(d*x + c) + 11)/(a^4*d*(sin(d*x + c) + 1)^3)
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^4\,d}+\frac {3\,{\sin \left (c+d\,x\right )}^2+\frac {9\,\sin \left (c+d\,x\right )}{2}+\frac {11}{6}}{a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3} \] Input:
int((cos(c + d*x)*sin(c + d*x)^3)/(a + a*sin(c + d*x))^4,x)
Output:
log(sin(c + d*x) + 1)/(a^4*d) + ((9*sin(c + d*x))/2 + 3*sin(c + d*x)^2 + 1 1/6)/(a^4*d*(sin(c + d*x) + 1)^3)
Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.52 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {6 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )^{3}+18 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )^{2}+18 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )+6 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right )-9 \sin \left (d x +c \right )^{3}-9 \sin \left (d x +c \right )^{2}+2}{6 a^{4} d \left (\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1\right )} \] Input:
int(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x)
Output:
(6*log(sin(c + d*x) + 1)*sin(c + d*x)**3 + 18*log(sin(c + d*x) + 1)*sin(c + d*x)**2 + 18*log(sin(c + d*x) + 1)*sin(c + d*x) + 6*log(sin(c + d*x) + 1 ) - 9*sin(c + d*x)**3 - 9*sin(c + d*x)**2 + 2)/(6*a**4*d*(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1))