Integrand size = 27, antiderivative size = 65 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \csc (c+d x)}{d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc ^3(c+d x)}{3 d}+\frac {a^3 \log (\sin (c+d x))}{d} \] Output:
-3*a^3*csc(d*x+c)/d-3/2*a^3*csc(d*x+c)^2/d-1/3*a^3*csc(d*x+c)^3/d+a^3*ln(s in(d*x+c))/d
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {3 \csc (c+d x)}{d}-\frac {3 \csc ^2(c+d x)}{2 d}-\frac {\csc ^3(c+d x)}{3 d}+\frac {\log (\sin (c+d x))}{d}\right ) \] Input:
Integrate[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
a^3*((-3*Csc[c + d*x])/d - (3*Csc[c + d*x]^2)/(2*d) - Csc[c + d*x]^3/(3*d) + Log[Sin[c + d*x]]/d)
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75, 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.
\(\displaystyle \int \cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x) (a \sin (c+d x)+a)^3}{\sin (c+d x)^4}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \csc ^4(c+d x) (\sin (c+d x) a+a)^3d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^3 \int \frac {\csc ^4(c+d x) (\sin (c+d x) a+a)^3}{a^4}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {a^3 \int \left (\frac {\csc ^4(c+d x)}{a}+\frac {3 \csc ^3(c+d x)}{a}+\frac {3 \csc ^2(c+d x)}{a}+\frac {\csc (c+d x)}{a}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \left (\log (a \sin (c+d x))-\frac {1}{3} \csc ^3(c+d x)-\frac {3}{2} \csc ^2(c+d x)-3 \csc (c+d x)\right )}{d}\) |
Input:
Int[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(-3*Csc[c + d*x] - (3*Csc[c + d*x]^2)/2 - Csc[c + d*x]^3/3 + Log[a*Si n[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 = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(-\frac {a^{3} \left (\frac {\csc \left (d x +c \right )^{3}}{3}+\frac {3 \csc \left (d x +c \right )^{2}}{2}+3 \csc \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )\right )\right )}{d}\) | \(45\) |
default | \(-\frac {a^{3} \left (\frac {\csc \left (d x +c \right )^{3}}{3}+\frac {3 \csc \left (d x +c \right )^{2}}{2}+3 \csc \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )\right )\right )}{d}\) | \(45\) |
risch | \(-i a^{3} x -\frac {2 i a^{3} c}{d}-\frac {2 i a^{3} \left (9 \,{\mathrm e}^{5 i \left (d x +c \right )}-22 \,{\mathrm e}^{3 i \left (d x +c \right )}+9 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}-9 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(118\) |
Input:
int(cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/d*a^3*(1/3*csc(d*x+c)^3+3/2*csc(d*x+c)^2+3*csc(d*x+c)+ln(csc(d*x+c)))
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.40 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {18 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \sin \left (d x + c\right ) - 20 \, a^{3} - 6 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/6*(18*a^3*cos(d*x + c)^2 - 9*a^3*sin(d*x + c) - 20*a^3 - 6*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*sin(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d) *sin(d*x + c))
\[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)*csc(d*x+c)**3*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cot(c + d*x)*csc(c + d*x)**3, x) + Integral(3*sin(c + d*x)* cot(c + d*x)*csc(c + d*x)**3, x) + Integral(3*sin(c + d*x)**2*cot(c + d*x) *csc(c + d*x)**3, x) + Integral(sin(c + d*x)**3*cot(c + d*x)*csc(c + d*x)* *3, x))
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - \frac {18 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/6*(6*a^3*log(sin(d*x + c)) - (18*a^3*sin(d*x + c)^2 + 9*a^3*sin(d*x + c) + 2*a^3)/sin(d*x + c)^3)/d
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {18 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/6*(6*a^3*log(abs(sin(d*x + c))) - (18*a^3*sin(d*x + c)^2 + 9*a^3*sin(d*x + c) + 2*a^3)/sin(d*x + c)^3)/d
Time = 18.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.26 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}\right )}{8\,d}-\frac {13\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int((cot(c + d*x)*(a + a*sin(c + d*x))^3)/sin(c + d*x)^3,x)
Output:
(a^3*log(tan(c/2 + (d*x)/2)))/d - (3*a^3*tan(c/2 + (d*x)/2)^2)/(8*d) - (a^ 3*tan(c/2 + (d*x)/2)^3)/(24*d) - (cot(c/2 + (d*x)/2)^3*(13*a^3*tan(c/2 + ( d*x)/2)^2 + a^3/3 + 3*a^3*tan(c/2 + (d*x)/2)))/(8*d) - (13*a^3*tan(c/2 + ( d*x)/2))/(8*d) - (a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{3}+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+9 \sin \left (d x +c \right )^{3}-36 \sin \left (d x +c \right )^{2}-18 \sin \left (d x +c \right )-4\right )}{12 \sin \left (d x +c \right )^{3} d} \] Input:
int(cot(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 12*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3 + 12*log(tan((c + d*x)/2))*sin(c + d*x)**3 + 9*sin(c + d*x)**3 - 36*sin(c + d*x)**2 - 18*s in(c + d*x) - 4))/(12*sin(c + d*x)**3*d)