Integrand size = 27, antiderivative size = 82 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (1+\sin (c+d x))}{a d} \] Output:
-csc(d*x+c)/a/d+1/2*csc(d*x+c)^2/a/d-1/3*csc(d*x+c)^3/a/d-ln(sin(d*x+c))/a /d+ln(1+sin(d*x+c))/a/d
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (1+\sin (c+d x))}{a d} \] Input:
Integrate[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
-(Csc[c + d*x]/(a*d)) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Sin[c + d*x]]/(a*d)
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95, 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, 54, 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.
\(\displaystyle \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^4 (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \frac {\csc ^4(c+d x)}{\sin (c+d x) a+a}d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^3 \int \frac {\csc ^4(c+d x)}{a^4 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {a^3 \int \left (\frac {\csc ^4(c+d x)}{a^5}-\frac {\csc ^3(c+d x)}{a^5}+\frac {\csc ^2(c+d x)}{a^5}-\frac {\csc (c+d x)}{a^5}+\frac {1}{a^4 (\sin (c+d x) a+a)}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \left (-\frac {\csc ^3(c+d x)}{3 a^4}+\frac {\csc ^2(c+d x)}{2 a^4}-\frac {\csc (c+d x)}{a^4}-\frac {\log (a \sin (c+d x))}{a^4}+\frac {\log (a \sin (c+d x)+a)}{a^4}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
(a^3*(-(Csc[c + d*x]/a^4) + Csc[c + d*x]^2/(2*a^4) - Csc[c + d*x]^3/(3*a^4 ) - Log[a*Sin[c + d*x]]/a^4 + Log[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[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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 = 0.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.57
method | result | size |
derivativedivides | \(-\frac {\frac {\csc \left (d x +c \right )^{3}}{3}-\frac {\csc \left (d x +c \right )^{2}}{2}+\csc \left (d x +c \right )-\ln \left (1+\csc \left (d x +c \right )\right )}{d a}\) | \(47\) |
default | \(-\frac {\frac {\csc \left (d x +c \right )^{3}}{3}-\frac {\csc \left (d x +c \right )^{2}}{2}+\csc \left (d x +c \right )-\ln \left (1+\csc \left (d x +c \right )\right )}{d a}\) | \(47\) |
risch | \(-\frac {2 i \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(123\) |
Input:
int(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/d/a*(1/3*csc(d*x+c)^3-1/2*csc(d*x+c)^2+csc(d*x+c)-ln(1+csc(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 6 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 8}{6 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/6*(6*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c))*sin(d*x + c) - 6*(cos(d *x + c)^2 - 1)*log(sin(d*x + c) + 1)*sin(d*x + c) + 6*cos(d*x + c)^2 + 3*s in(d*x + c) - 8)/((a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))
\[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)*csc(c + d*x)**3/(sin(c + d*x) + 1), x)/a
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
1/6*(6*log(sin(d*x + c) + 1)/a - 6*log(sin(d*x + c))/a - (6*sin(d*x + c)^2 - 3*sin(d*x + c) + 2)/(a*sin(d*x + c)^3))/d
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a d} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a d} - \frac {6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{6 \, a d \sin \left (d x + c\right )^{3}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
log(abs(sin(d*x + c) + 1))/(a*d) - log(abs(sin(d*x + c)))/(a*d) - 1/6*(6*s in(d*x + c)^2 - 3*sin(d*x + c) + 2)/(a*d*sin(d*x + c)^3)
Time = 18.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.70 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a\,d}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{3}\right )}{8\,a\,d} \] Input:
int(cot(c + d*x)/(sin(c + d*x)^3*(a + a*sin(c + d*x))),x)
Output:
tan(c/2 + (d*x)/2)^2/(8*a*d) - tan(c/2 + (d*x)/2)^3/(24*a*d) - log(tan(c/2 + (d*x)/2))/(a*d) + (2*log(tan(c/2 + (d*x)/2) + 1))/(a*d) - (5*tan(c/2 + (d*x)/2))/(8*a*d) - (cot(c/2 + (d*x)/2)^3*(5*tan(c/2 + (d*x)/2)^2 - tan(c/ 2 + (d*x)/2) + 1/3))/(8*a*d)
Time = 0.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3}-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}-3 \sin \left (d x +c \right )^{3}-12 \sin \left (d x +c \right )^{2}+6 \sin \left (d x +c \right )-4}{12 \sin \left (d x +c \right )^{3} a d} \] Input:
int(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x)
Output:
(24*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 - 12*log(tan((c + d*x)/2))*s in(c + d*x)**3 - 3*sin(c + d*x)**3 - 12*sin(c + d*x)**2 + 6*sin(c + d*x) - 4)/(12*sin(c + d*x)**3*a*d)