Integrand size = 27, antiderivative size = 110 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{d}-\frac {a^2 \csc ^3(c+d x)}{3 d}-\frac {4 a^2 \log (\sin (c+d x))}{d}-\frac {a^2 \sin (c+d x)}{d}+\frac {a^2 \sin ^2(c+d x)}{d}+\frac {a^2 \sin ^3(c+d x)}{3 d} \] Output:
a^2*csc(d*x+c)/d-a^2*csc(d*x+c)^2/d-1/3*a^2*csc(d*x+c)^3/d-4*a^2*ln(sin(d* x+c))/d-a^2*sin(d*x+c)/d+a^2*sin(d*x+c)^2/d+1/3*a^2*sin(d*x+c)^3/d
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (3 \csc (c+d x)-3 \csc ^2(c+d x)-\csc ^3(c+d x)-12 \log (\sin (c+d x))-3 \sin (c+d x)+3 \sin ^2(c+d x)+\sin ^3(c+d x)\right )}{3 d} \] Input:
Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]
Output:
(a^2*(3*Csc[c + d*x] - 3*Csc[c + d*x]^2 - Csc[c + d*x]^3 - 12*Log[Sin[c + d*x]] - 3*Sin[c + d*x] + 3*Sin[c + d*x]^2 + Sin[c + d*x]^3))/(3*d)
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89, 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, 99, 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.
| \(\displaystyle \int \cos (c+d x) \cot ^4(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)^2}{\sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \csc ^4(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^4d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^4}{a^4}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^2 \csc ^4(c+d x)+2 a^2 \csc ^3(c+d x)-a^2 \csc ^2(c+d x)-4 a^2 \csc (c+d x)-a^2+a^2 \sin ^2(c+d x)+2 a^2 \sin (c+d x)\right )d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{3} a^3 \sin ^3(c+d x)+a^3 \sin ^2(c+d x)-a^3 \sin (c+d x)-\frac {1}{3} a^3 \csc ^3(c+d x)-a^3 \csc ^2(c+d x)+a^3 \csc (c+d x)-4 a^3 \log (a \sin (c+d x))}{a d}\) |
Input:
Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]
Output:
(a^3*Csc[c + d*x] - a^3*Csc[c + d*x]^2 - (a^3*Csc[c + d*x]^3)/3 - 4*a^3*Lo g[a*Sin[c + d*x]] - a^3*Sin[c + d*x] + a^3*Sin[c + d*x]^2 + (a^3*Sin[c + d *x]^3)/3)/(a*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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 2.56 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(177\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(177\) |
| risch | \(4 i a^{2} x +\frac {i a^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {3 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}-\frac {i a^{2} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {8 i a^{2} c}{d}+\frac {2 i a^{2} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-2 \,{\mathrm e}^{3 i \left (d x +c \right )}-6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+6 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {4 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(225\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-1/sin(d*x+c)*cos(d*x+c)^6-(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*s in(d*x+c))+2*a^2*(-1/2/sin(d*x+c)^2*cos(d*x+c)^6-1/2*cos(d*x+c)^4-cos(d*x+ c)^2-2*ln(sin(d*x+c)))+a^2*(-1/3/sin(d*x+c)^3*cos(d*x+c)^6+1/sin(d*x+c)*co s(d*x+c)^6+(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {2 \, a^{2} \cos \left (d x + c\right )^{6} - 24 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 3 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/6*(2*a^2*cos(d*x + c)^6 - 24*(a^2*cos(d*x + c)^2 - a^2)*log(1/2*sin(d*x + c))*sin(d*x + c) - 3*(2*a^2*cos(d*x + c)^4 - 3*a^2*cos(d*x + c)^2 - a^2) *sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))
\[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int 2 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**4*(a+a*sin(d*x+c))**2,x)
Output:
a**2*(Integral(cos(c + d*x)*cot(c + d*x)**4, x) + Integral(2*sin(c + d*x)* cos(c + d*x)*cot(c + d*x)**4, x) + Integral(sin(c + d*x)**2*cos(c + d*x)*c ot(c + d*x)**4, x))
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \sin \left (d x + c\right )^{3} + 3 \, a^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) - 3 \, a^{2} \sin \left (d x + c\right ) + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, a^{2} \sin \left (d x + c\right ) - a^{2}}{\sin \left (d x + c\right )^{3}}}{3 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
1/3*(a^2*sin(d*x + c)^3 + 3*a^2*sin(d*x + c)^2 - 12*a^2*log(sin(d*x + c)) - 3*a^2*sin(d*x + c) + (3*a^2*sin(d*x + c)^2 - 3*a^2*sin(d*x + c) - a^2)/s in(d*x + c)^3)/d
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \sin \left (d x + c\right )^{3} + 3 \, a^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 3 \, a^{2} \sin \left (d x + c\right ) + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, a^{2} \sin \left (d x + c\right ) - a^{2}}{\sin \left (d x + c\right )^{3}}}{3 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/3*(a^2*sin(d*x + c)^3 + 3*a^2*sin(d*x + c)^2 - 12*a^2*log(abs(sin(d*x + c))) - 3*a^2*sin(d*x + c) + (3*a^2*sin(d*x + c)^2 - 3*a^2*sin(d*x + c) - a ^2)/sin(d*x + c)^3)/d
Time = 18.64 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.62 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {4\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-30\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-26\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^2}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {4\,a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int(cos(c + d*x)*cot(c + d*x)^4*(a + a*sin(c + d*x))^2,x)
Output:
(3*a^2*tan(c/2 + (d*x)/2))/(8*d) - (a^2*tan(c/2 + (d*x)/2)^3)/(24*d) - (4* a^2*log(tan(c/2 + (d*x)/2)))/d - (a^2*tan(c/2 + (d*x)/2)^2)/(4*d) - (6*a^2 *tan(c/2 + (d*x)/2)^3 - 2*a^2*tan(c/2 + (d*x)/2)^2 + 8*a^2*tan(c/2 + (d*x) /2)^4 - 26*a^2*tan(c/2 + (d*x)/2)^5 + 2*a^2*tan(c/2 + (d*x)/2)^6 - 30*a^2* tan(c/2 + (d*x)/2)^7 + 13*a^2*tan(c/2 + (d*x)/2)^8 + a^2/3 + 2*a^2*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 24*tan(c/2 + (d*x)/2)^5 + 24*tan (c/2 + (d*x)/2)^7 + 8*tan(c/2 + (d*x)/2)^9)) + (4*a^2*log(tan(c/2 + (d*x)/ 2)^2 + 1))/d
Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{3}-48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+4 \sin \left (d x +c \right )^{6}+12 \sin \left (d x +c \right )^{5}-12 \sin \left (d x +c \right )^{4}+15 \sin \left (d x +c \right )^{3}+12 \sin \left (d x +c \right )^{2}-12 \sin \left (d x +c \right )-4\right )}{12 \sin \left (d x +c \right )^{3} d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^4*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(48*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3 - 48*log(tan((c + d *x)/2))*sin(c + d*x)**3 + 4*sin(c + d*x)**6 + 12*sin(c + d*x)**5 - 12*sin( c + d*x)**4 + 15*sin(c + d*x)**3 + 12*sin(c + d*x)**2 - 12*sin(c + d*x) - 4))/(12*sin(c + d*x)**3*d)