Integrand size = 19, antiderivative size = 74 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\log (\sin (c+d x))}{a^3 d}-\frac {\log (1+\sin (c+d x))}{a^3 d}+\frac {1}{2 a d (a+a \sin (c+d x))^2}+\frac {1}{d \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
ln(sin(d*x+c))/a^3/d-ln(1+sin(d*x+c))/a^3/d+1/2/a/d/(a+a*sin(d*x+c))^2+1/d /(a^3+a^3*sin(d*x+c))
Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.70 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \log (\sin (c+d x))-2 \log (1+\sin (c+d x))+\frac {3+2 \sin (c+d x)}{(1+\sin (c+d x))^2}}{2 a^3 d} \] Input:
Integrate[Cot[c + d*x]/(a + a*Sin[c + d*x])^3,x]
Output:
(2*Log[Sin[c + d*x]] - 2*Log[1 + Sin[c + d*x]] + (3 + 2*Sin[c + d*x])/(1 + Sin[c + d*x])^2)/(2*a^3*d)
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3186, 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.
\(\displaystyle \int \frac {\cot (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x) (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3186 |
\(\displaystyle \frac {\int \frac {\csc (c+d x)}{a (\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \left (\frac {\csc (c+d x)}{a^4}-\frac {1}{a^3 (\sin (c+d x) a+a)}-\frac {1}{a^2 (\sin (c+d x) a+a)^2}-\frac {1}{a (\sin (c+d x) a+a)^3}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\log (a \sin (c+d x))}{a^3}-\frac {\log (a \sin (c+d x)+a)}{a^3}+\frac {1}{a^2 (a \sin (c+d x)+a)}+\frac {1}{2 a (a \sin (c+d x)+a)^2}}{d}\) |
Input:
Int[Cot[c + d*x]/(a + a*Sin[c + d*x])^3,x]
Output:
(Log[a*Sin[c + d*x]]/a^3 - Log[a + a*Sin[c + d*x]]/a^3 + 1/(2*a*(a + a*Sin [c + d*x])^2) + 1/(a^2*(a + a*Sin[c + d*x])))/d
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) ^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Time = 1.90 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {\ln \left (\sin \left (d x +c \right )\right )+\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{1+\sin \left (d x +c \right )}-\ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(49\) |
default | \(\frac {\ln \left (\sin \left (d x +c \right )\right )+\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{1+\sin \left (d x +c \right )}-\ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(49\) |
risch | \(\frac {2 i \left (-{\mathrm e}^{i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{3 i \left (d x +c \right )}\right )}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4}}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(98\) |
Input:
int(cot(d*x+c)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(ln(sin(d*x+c))+1/2/(1+sin(d*x+c))^2+1/(1+sin(d*x+c))-ln(1+sin(d*x +c)))
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, \sin \left (d x + c\right ) - 3}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \] Input:
integrate(cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/2*(2*(cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(1/2*sin(d*x + c)) - 2*(co s(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(sin(d*x + c) + 1) - 2*sin(d*x + c) - 3)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)
\[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cot {\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x ) + 1), x)/a**3
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {2 \, \sin \left (d x + c\right ) + 3}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \] Input:
integrate(cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*((2*sin(d*x + c) + 3)/(a^3*sin(d*x + c)^2 + 2*a^3*sin(d*x + c) + a^3) - 2*log(sin(d*x + c) + 1)/a^3 + 2*log(sin(d*x + c))/a^3)/d
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} + \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3} d} + \frac {2 \, \sin \left (d x + c\right ) + 3}{2 \, a^{3} d {\left (\sin \left (d x + c\right ) + 1\right )}^{2}} \] Input:
integrate(cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-log(abs(sin(d*x + c) + 1))/(a^3*d) + log(abs(sin(d*x + c)))/(a^3*d) + 1/2 *(2*sin(d*x + c) + 3)/(a^3*d*(sin(d*x + c) + 1)^2)
Time = 17.89 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.00 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a^3\right )}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d} \] Input:
int(cot(c + d*x)/(a + a*sin(c + d*x))^3,x)
Output:
log(tan(c/2 + (d*x)/2))/(a^3*d) - (4*tan(c/2 + (d*x)/2) + 6*tan(c/2 + (d*x )/2)^2 + 4*tan(c/2 + (d*x)/2)^3)/(d*(6*a^3*tan(c/2 + (d*x)/2)^2 + 4*a^3*ta n(c/2 + (d*x)/2)^3 + a^3*tan(c/2 + (d*x)/2)^4 + a^3 + 4*a^3*tan(c/2 + (d*x )/2))) - (2*log(tan(c/2 + (d*x)/2) + 1))/(a^3*d)
Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.97 \[ \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}-8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sin \left (d x +c \right )^{2}+2}{2 a^{3} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int(cot(d*x+c)/(a+a*sin(d*x+c))^3,x)
Output:
( - 4*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 - 8*log(tan((c + d*x)/2) + 1)*sin(c + d*x) - 4*log(tan((c + d*x)/2) + 1) + 2*log(tan((c + d*x)/2))*s in(c + d*x)**2 + 4*log(tan((c + d*x)/2))*sin(c + d*x) + 2*log(tan((c + d*x )/2)) - sin(c + d*x)**2 + 2)/(2*a**3*d*(sin(c + d*x)**2 + 2*sin(c + d*x) + 1))