Integrand size = 27, antiderivative size = 61 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}+\frac {3 a^3 \log (\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d} \] Output:
-3*a^3*csc(d*x+c)/d-1/2*a^3*csc(d*x+c)^2/d+3*a^3*ln(sin(d*x+c))/d+a^3*sin( d*x+c)/d
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {3 \csc (c+d x)}{d}-\frac {\csc ^2(c+d x)}{2 d}+\frac {3 \log (\sin (c+d x))}{d}+\frac {\sin (c+d x)}{d}\right ) \] Input:
Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
a^3*((-3*Csc[c + d*x])/d - Csc[c + d*x]^2/(2*d) + (3*Log[Sin[c + d*x]])/d + Sin[c + d*x]/d)
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82, 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.
\(\displaystyle \int \cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x) (a \sin (c+d x)+a)^3}{\sin (c+d x)^3}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \csc ^3(c+d x) (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \int \frac {\csc ^3(c+d x) (\sin (c+d x) a+a)^3}{a^3}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {a^2 \int \left (\csc ^3(c+d x)+3 \csc ^2(c+d x)+3 \csc (c+d x)+1\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \left (a \sin (c+d x)-\frac {1}{2} a \csc ^2(c+d x)-3 a \csc (c+d x)+3 a \log (a \sin (c+d x))\right )}{d}\) |
Input:
Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(a^2*(-3*a*Csc[c + d*x] - (a*Csc[c + d*x]^2)/2 + 3*a*Log[a*Sin[c + d*x]] + a*Sin[c + d*x]))/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 = 0.71 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {a^{3} \left (\frac {\csc \left (d x +c \right )^{2}}{2}+3 \csc \left (d x +c \right )+3 \ln \left (\csc \left (d x +c \right )\right )-\frac {1}{\csc \left (d x +c \right )}\right )}{d}\) | \(47\) |
default | \(-\frac {a^{3} \left (\frac {\csc \left (d x +c \right )^{2}}{2}+3 \csc \left (d x +c \right )+3 \ln \left (\csc \left (d x +c \right )\right )-\frac {1}{\csc \left (d x +c \right )}\right )}{d}\) | \(47\) |
risch | \(-3 i a^{3} x -\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {6 i a^{3} c}{d}-\frac {2 i a^{3} \left (i {\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(132\) |
Input:
int(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/d*a^3*(1/2*csc(d*x+c)^2+3*csc(d*x+c)+3*ln(csc(d*x+c))-1/csc(d*x+c))
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} + 6 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/2*(a^3 + 6*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*sin(d*x + c)) + 2*(a^3*cos (d*x + c)^2 + 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)
\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cot {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cot {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cot {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)*csc(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cot(c + d*x)*csc(c + d*x)**2, x) + Integral(3*sin(c + d*x)* cot(c + d*x)*csc(c + d*x)**2, x) + Integral(3*sin(c + d*x)**2*cot(c + d*x) *csc(c + d*x)**2, x) + Integral(sin(c + d*x)**3*cot(c + d*x)*csc(c + d*x)* *2, x))
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 2 \, a^{3} \sin \left (d x + c\right ) - \frac {6 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/2*(6*a^3*log(sin(d*x + c)) + 2*a^3*sin(d*x + c) - (6*a^3*sin(d*x + c) + a^3)/sin(d*x + c)^2)/d
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 2 \, a^{3} \sin \left (d x + c\right ) - \frac {6 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/2*(6*a^3*log(abs(sin(d*x + c))) + 2*a^3*sin(d*x + c) - (6*a^3*sin(d*x + c) + a^3)/sin(d*x + c)^2)/d
Time = 18.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.67 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {3\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int((cot(c + d*x)*(a + a*sin(c + d*x))^3)/sin(c + d*x)^2,x)
Output:
(3*a^3*log(tan(c/2 + (d*x)/2)))/d - ((a^3*tan(c/2 + (d*x)/2)^2)/2 - 2*a^3* tan(c/2 + (d*x)/2)^3 + a^3/2 + 6*a^3*tan(c/2 + (d*x)/2))/(d*(4*tan(c/2 + ( d*x)/2)^2 + 4*tan(c/2 + (d*x)/2)^4)) - (a^3*tan(c/2 + (d*x)/2)^2)/(8*d) - (3*a^3*tan(c/2 + (d*x)/2))/(2*d) - (3*a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.48 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+8 \sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}-24 \sin \left (d x +c \right )-4\right )}{8 \sin \left (d x +c \right )^{2} d} \] Input:
int(cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 24*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2 + 24*log(tan((c + d*x)/2))*sin(c + d*x)**2 + 8*sin(c + d*x)**3 + 3*sin(c + d*x)**2 - 24*si n(c + d*x) - 4))/(8*sin(c + d*x)**2*d)