Integrand size = 19, antiderivative size = 75 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\log (\sin (c+d x))}{a^3 d}-\frac {\log (a+b \sin (c+d x))}{a^3 d}+\frac {1}{2 a d (a+b \sin (c+d x))^2}+\frac {1}{a^2 d (a+b \sin (c+d x))} \] Output:
ln(sin(d*x+c))/a^3/d-ln(a+b*sin(d*x+c))/a^3/d+1/2/a/d/(a+b*sin(d*x+c))^2+1 /a^2/d/(a+b*sin(d*x+c))
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {2 \log (\sin (c+d x))-2 \log (a+b \sin (c+d x))+\frac {a (3 a+2 b \sin (c+d x))}{(a+b \sin (c+d x))^2}}{2 a^3 d} \] Input:
Integrate[Cot[c + d*x]/(a + b*Sin[c + d*x])^3,x]
Output:
(2*Log[Sin[c + d*x]] - 2*Log[a + b*Sin[c + d*x]] + (a*(3*a + 2*b*Sin[c + d *x]))/(a + b*Sin[c + d*x])^2)/(2*a^3*d)
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3200, 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.
\(\displaystyle \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x) (a+b \sin (c+d x))^3}dx\) |
\(\Big \downarrow \) 3200 |
\(\displaystyle \frac {\int \frac {\csc (c+d x)}{b (a+b \sin (c+d x))^3}d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \left (\frac {\csc (c+d x)}{a^3 b}-\frac {1}{a^3 (a+b \sin (c+d x))}-\frac {1}{a^2 (a+b \sin (c+d x))^2}-\frac {1}{a (a+b \sin (c+d x))^3}\right )d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\log (b \sin (c+d x))}{a^3}-\frac {\log (a+b \sin (c+d x))}{a^3}+\frac {1}{a^2 (a+b \sin (c+d x))}+\frac {1}{2 a (a+b \sin (c+d x))^2}}{d}\) |
Input:
Int[Cot[c + d*x]/(a + b*Sin[c + d*x])^3,x]
Output:
(Log[b*Sin[c + d*x]]/a^3 - Log[a + b*Sin[c + d*x]]/a^3 + 1/(2*a*(a + b*Sin [c + d*x])^2) + 1/(a^2*(a + b*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)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 1.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{3}}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{2 a \left (a +b \sin \left (d x +c \right )\right )^{2}}}{d}\) | \(66\) |
default | \(\frac {\frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{3}}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{2 a \left (a +b \sin \left (d x +c \right )\right )^{2}}}{d}\) | \(66\) |
risch | \(\frac {2 i \left (3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )} b \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )} b -b +2 i {\mathrm e}^{i \left (d x +c \right )} a \right )^{2} a^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(133\) |
Input:
int(cot(d*x+c)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/a^3*ln(sin(d*x+c))-1/a^3*ln(a+b*sin(d*x+c))+1/a^2/(a+b*sin(d*x+c))+ 1/2/a/(a+b*sin(d*x+c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (73) = 146\).
Time = 0.15 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.05 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {2 \, a b \sin \left (d x + c\right ) + 3 \, a^{2} + 2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{2 \, {\left (a^{3} b^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{4} b d \sin \left (d x + c\right ) - {\left (a^{5} + a^{3} b^{2}\right )} d\right )}} \] Input:
integrate(cot(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-1/2*(2*a*b*sin(d*x + c) + 3*a^2 + 2*(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)*log(b*sin(d*x + c) + a) - 2*(b^2*cos(d*x + c)^2 - 2*a*b*s in(d*x + c) - a^2 - b^2)*log(-1/2*sin(d*x + c)))/(a^3*b^2*d*cos(d*x + c)^2 - 2*a^4*b*d*sin(d*x + c) - (a^5 + a^3*b^2)*d)
\[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cot(d*x+c)/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)/(a + b*sin(c + d*x))**3, x)
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \, b \sin \left (d x + c\right ) + 3 \, a}{a^{2} b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{3} b \sin \left (d x + c\right ) + a^{4}} - \frac {2 \, \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{3}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \] Input:
integrate(cot(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*((2*b*sin(d*x + c) + 3*a)/(a^2*b^2*sin(d*x + c)^2 + 2*a^3*b*sin(d*x + c) + a^4) - 2*log(b*sin(d*x + c) + a)/a^3 + 2*log(sin(d*x + c))/a^3)/d
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{3} d} + \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3} d} + \frac {2 \, a b \sin \left (d x + c\right ) + 3 \, a^{2}}{2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{2} a^{3} d} \] Input:
integrate(cot(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
-log(abs(b*sin(d*x + c) + a))/(a^3*d) + log(abs(sin(d*x + c)))/(a^3*d) + 1 /2*(2*a*b*sin(d*x + c) + 3*a^2)/((b*sin(d*x + c) + a)^2*a^3*d)
Time = 17.51 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.92 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{a^3\,d}-\frac {6\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d\,\left (a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^5+4\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,a^3\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4+4\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,a^2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4+4\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,a^2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \] Input:
int(cot(c + d*x)/(a + b*sin(c + d*x))^3,x)
Output:
log(tan(c/2 + (d*x)/2))/(a^3*d) - log(a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c /2 + (d*x)/2)^2)/(a^3*d) - (6*b^2*tan(c/2 + (d*x)/2)^2)/(d*(2*a^5*tan(c/2 + (d*x)/2)^2 + a^5*tan(c/2 + (d*x)/2)^4 + a^5 + 4*a^3*b^2*tan(c/2 + (d*x)/ 2)^2 + 4*a^4*b*tan(c/2 + (d*x)/2) + 4*a^4*b*tan(c/2 + (d*x)/2)^3)) - (4*b* tan(c/2 + (d*x)/2))/(d*(2*a^4*tan(c/2 + (d*x)/2)^2 + a^4*tan(c/2 + (d*x)/2 )^4 + a^4 + 4*a^2*b^2*tan(c/2 + (d*x)/2)^2 + 4*a^3*b*tan(c/2 + (d*x)/2) + 4*a^3*b*tan(c/2 + (d*x)/2)^3)) - (4*b*tan(c/2 + (d*x)/2)^3)/(d*(2*a^4*tan( c/2 + (d*x)/2)^2 + a^4*tan(c/2 + (d*x)/2)^4 + a^4 + 4*a^2*b^2*tan(c/2 + (d *x)/2)^2 + 4*a^3*b*tan(c/2 + (d*x)/2) + 4*a^3*b*tan(c/2 + (d*x)/2)^3))
Time = 0.19 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.00 \[ \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right )^{2} b^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right ) a b -2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} b^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a b +2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-\sin \left (d x +c \right )^{2} b^{2}+2 a^{2}}{2 a^{3} d \left (\sin \left (d x +c \right )^{2} b^{2}+2 \sin \left (d x +c \right ) a b +a^{2}\right )} \] Input:
int(cot(d*x+c)/(a+b*sin(d*x+c))^3,x)
Output:
( - 2*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d*x)** 2*b**2 - 4*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d *x)*a*b - 2*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*a**2 + 2 *log(tan((c + d*x)/2))*sin(c + d*x)**2*b**2 + 4*log(tan((c + d*x)/2))*sin( c + d*x)*a*b + 2*log(tan((c + d*x)/2))*a**2 - sin(c + d*x)**2*b**2 + 2*a** 2)/(2*a**3*d*(sin(c + d*x)**2*b**2 + 2*sin(c + d*x)*a*b + a**2))