Integrand size = 27, antiderivative size = 93 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6 \log (1+\sin (c+d x))}{a^3 d}-\frac {3 \sin (c+d x)}{a^3 d}+\frac {\sin ^2(c+d x)}{2 a^3 d}-\frac {1}{2 a d (a+a \sin (c+d x))^2}+\frac {4}{d \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
6*ln(1+sin(d*x+c))/a^3/d-3*sin(d*x+c)/a^3/d+1/2*sin(d*x+c)^2/a^3/d-1/2/a/d /(a+a*sin(d*x+c))^2+4/d/(a^3+a^3*sin(d*x+c))
Time = 0.33 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {7+12 \log (1+\sin (c+d x))+(2+24 \log (1+\sin (c+d x))) \sin (c+d x)+(-11+12 \log (1+\sin (c+d x))) \sin ^2(c+d x)-4 \sin ^3(c+d x)+\sin ^4(c+d x)}{2 a^3 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]
Output:
(7 + 12*Log[1 + Sin[c + d*x]] + (2 + 24*Log[1 + Sin[c + d*x]])*Sin[c + d*x ] + (-11 + 12*Log[1 + Sin[c + d*x]])*Sin[c + d*x]^2 - 4*Sin[c + d*x]^3 + S in[c + d*x]^4)/(2*a^3*d*(1 + Sin[c + d*x])^2)
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \cos (c+d x)}{(a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {\sin ^4(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^4 \sin ^4(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\int \left (\frac {a^4}{(\sin (c+d x) a+a)^3}-\frac {4 a^3}{(\sin (c+d x) a+a)^2}+\frac {6 a^2}{\sin (c+d x) a+a}+\sin (c+d x) a-3 a\right )d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^4}{2 (a \sin (c+d x)+a)^2}+\frac {4 a^3}{a \sin (c+d x)+a}+\frac {1}{2} a^2 \sin ^2(c+d x)-3 a^2 \sin (c+d x)+6 a^2 \log (a \sin (c+d x)+a)}{a^5 d}\) |
Input:
Int[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]
Output:
(6*a^2*Log[a + a*Sin[c + d*x]] - 3*a^2*Sin[c + d*x] + (a^2*Sin[c + d*x]^2) /2 - a^4/(2*(a + a*Sin[c + d*x])^2) + (4*a^3)/(a + a*Sin[c + d*x]))/(a^5*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 = 0.80 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(\frac {\frac {\sin \left (d x +c \right )^{2}}{2}-3 \sin \left (d x +c \right )+6 \ln \left (1+\sin \left (d x +c \right )\right )+\frac {4}{1+\sin \left (d x +c \right )}-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{3}}\) | \(62\) |
| default | \(\frac {\frac {\sin \left (d x +c \right )^{2}}{2}-3 \sin \left (d x +c \right )+6 \ln \left (1+\sin \left (d x +c \right )\right )+\frac {4}{1+\sin \left (d x +c \right )}-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{3}}\) | \(62\) |
| parallelrisch | \(\frac {\left (-48 \cos \left (2 d x +2 c \right )+192 \sin \left (d x +c \right )+144\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (96 \cos \left (2 d x +2 c \right )-384 \sin \left (d x +c \right )-288\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-68 \cos \left (2 d x +2 c \right )-\cos \left (4 d x +4 c \right )+120 \sin \left (d x +c \right )-8 \sin \left (3 d x +3 c \right )+69}{8 d \,a^{3} \left (-3+\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )\right )}\) | \(141\) |
| risch | \(-\frac {6 i x}{a^{3}}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {12 i c}{d \,a^{3}}+\frac {2 i \left (7 i {\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4}}+\frac {12 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}\) | \(168\) |
| norman | \(\frac {-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d a}-\frac {124 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d a}-\frac {124 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d a}-\frac {416 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d a}-\frac {416 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}-\frac {672 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d a}-\frac {672 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}-\frac {260 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d a}-\frac {260 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d a}-\frac {48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}-\frac {48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d a}-\frac {580 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d a}-\frac {580 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{3}}-\frac {6 \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \,a^{3}}\) | \(341\) |
Input:
int(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(1/2*sin(d*x+c)^2-3*sin(d*x+c)+6*ln(1+sin(d*x+c))+4/(1+sin(d*x+c)) -1/2/(1+sin(d*x+c))^2)
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.15 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{2} - 24 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) - 8}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/4*(2*cos(d*x + c)^4 + 19*cos(d*x + c)^2 - 24*(cos(d*x + c)^2 - 2*sin(d* x + c) - 2)*log(sin(d*x + c) + 1) + 2*(4*cos(d*x + c)^2 - 3)*sin(d*x + c) - 8)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (80) = 160\).
Time = 1.00 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.73 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\begin {cases} \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {24 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {\sin ^{4}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} - \frac {4 \sin ^{3}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {24 \sin {\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} + \frac {18}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)**4/(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((12*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(2*a**3*d*sin(c + d*x) **2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) + 24*log(sin(c + d*x) + 1)*sin(c + d*x)/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d) + 12*l og(sin(c + d*x) + 1)/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2 *a**3*d) + sin(c + d*x)**4/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d* x) + 2*a**3*d) - 4*sin(c + d*x)**3/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*si n(c + d*x) + 2*a**3*d) + 24*sin(c + d*x)/(2*a**3*d*sin(c + d*x)**2 + 4*a** 3*d*sin(c + d*x) + 2*a**3*d) + 18/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin (c + d*x) + 2*a**3*d), Ne(d, 0)), (x*sin(c)**4*cos(c)/(a*sin(c) + a)**3, T rue))
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {8 \, \sin \left (d x + c\right ) + 7}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} + \frac {\sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a^{3}} + \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/2*((8*sin(d*x + c) + 7)/(a^3*sin(d*x + c)^2 + 2*a^3*sin(d*x + c) + a^3) + (sin(d*x + c)^2 - 6*sin(d*x + c))/a^3 + 12*log(sin(d*x + c) + 1)/a^3)/d
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} + \frac {8 \, \sin \left (d x + c\right ) + 7}{2 \, a^{3} d {\left (\sin \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} d \sin \left (d x + c\right )^{2} - 6 \, a^{3} d \sin \left (d x + c\right )}{2 \, a^{6} d^{2}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
6*log(abs(sin(d*x + c) + 1))/(a^3*d) + 1/2*(8*sin(d*x + c) + 7)/(a^3*d*(si n(d*x + c) + 1)^2) + 1/2*(a^3*d*sin(d*x + c)^2 - 6*a^3*d*sin(d*x + c))/(a^ 6*d^2)
Time = 18.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^3\,d}+\frac {4\,\sin \left (c+d\,x\right )+\frac {7}{2}}{d\,\left (a^3\,{\sin \left (c+d\,x\right )}^2+2\,a^3\,\sin \left (c+d\,x\right )+a^3\right )}-\frac {3\,\sin \left (c+d\,x\right )}{a^3\,d}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a^3\,d} \] Input:
int((cos(c + d*x)*sin(c + d*x)^4)/(a + a*sin(c + d*x))^3,x)
Output:
(6*log(sin(c + d*x) + 1))/(a^3*d) + (4*sin(c + d*x) + 7/2)/(d*(2*a^3*sin(c + d*x) + a^3 + a^3*sin(c + d*x)^2)) - (3*sin(c + d*x))/(a^3*d) + sin(c + d*x)^2/(2*a^3*d)
Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {12 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )^{2}+24 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )+12 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right )+\sin \left (d x +c \right )^{4}-4 \sin \left (d x +c \right )^{3}-12 \sin \left (d x +c \right )^{2}+6}{2 a^{3} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int(cos(d*x+c)*sin(d*x+c)^4/(a+a*sin(d*x+c))^3,x)
Output:
(12*log(sin(c + d*x) + 1)*sin(c + d*x)**2 + 24*log(sin(c + d*x) + 1)*sin(c + d*x) + 12*log(sin(c + d*x) + 1) + sin(c + d*x)**4 - 4*sin(c + d*x)**3 - 12*sin(c + d*x)**2 + 6)/(2*a**3*d*(sin(c + d*x)**2 + 2*sin(c + d*x) + 1))