Integrand size = 25, antiderivative size = 68 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc (c+d x)}{a^2 d}-\frac {2 \log (\sin (c+d x))}{a^2 d}+\frac {2 \log (1+\sin (c+d x))}{a^2 d}-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )} \] Output:
-csc(d*x+c)/a^2/d-2*ln(sin(d*x+c))/a^2/d+2*ln(1+sin(d*x+c))/a^2/d-1/d/(a^2 +a^2*sin(d*x+c))
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.66 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc (c+d x)+2 \log (\sin (c+d x))-2 \log (1+\sin (c+d x))+\frac {1}{1+\sin (c+d x)}}{a^2 d} \] Input:
Integrate[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]
Output:
-((Csc[c + d*x] + 2*Log[Sin[c + d*x]] - 2*Log[1 + Sin[c + d*x]] + (1 + Sin [c + d*x])^(-1))/(a^2*d))
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int \frac {\cot (c+d x) \csc (c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^2 (a \sin (c+d x)+a)^2}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \frac {\csc ^2(c+d x)}{(\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {\csc ^2(c+d x)}{a^2 (\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {a \int \left (\frac {\csc ^2(c+d x)}{a^4}-\frac {2 \csc (c+d x)}{a^4}+\frac {2}{a^3 (\sin (c+d x) a+a)}+\frac {1}{a^2 (\sin (c+d x) a+a)^2}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (-\frac {\csc (c+d x)}{a^3}-\frac {2 \log (a \sin (c+d x))}{a^3}+\frac {2 \log (a \sin (c+d x)+a)}{a^3}-\frac {1}{a^2 (a \sin (c+d x)+a)}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]
Output:
(a*(-(Csc[c + d*x]/a^3) - (2*Log[a*Sin[c + d*x]])/a^3 + (2*Log[a + a*Sin[c + d*x]])/a^3 - 1/(a^2*(a + 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[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 = 0.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
derivativedivides | \(-\frac {\csc \left (d x +c \right )-\frac {1}{1+\csc \left (d x +c \right )}-2 \ln \left (1+\csc \left (d x +c \right )\right )}{d \,a^{2}}\) | \(39\) |
default | \(-\frac {\csc \left (d x +c \right )-\frac {1}{1+\csc \left (d x +c \right )}-2 \ln \left (1+\csc \left (d x +c \right )\right )}{d \,a^{2}}\) | \(39\) |
risch | \(-\frac {4 i \left (i {\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} d \,a^{2}}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{2}}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{2}}\) | \(112\) |
Input:
int(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/d/a^2*(csc(d*x+c)-1/(1+csc(d*x+c))-2*ln(1+csc(d*x+c)))
Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, \sin \left (d x + c\right ) - 1}{a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \sin \left (d x + c\right ) - a^{2} d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Output:
-(2*(cos(d*x + c)^2 - sin(d*x + c) - 1)*log(1/2*sin(d*x + c)) - 2*(cos(d*x + c)^2 - sin(d*x + c) - 1)*log(sin(d*x + c) + 1) - 2*sin(d*x + c) - 1)/(a ^2*d*cos(d*x + c)^2 - a^2*d*sin(d*x + c) - a^2*d)
\[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cot {\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)*csc(c + d*x)/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, \sin \left (d x + c\right ) + 1}{a^{2} \sin \left (d x + c\right )^{2} + a^{2} \sin \left (d x + c\right )} - \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
Output:
-((2*sin(d*x + c) + 1)/(a^2*sin(d*x + c)^2 + a^2*sin(d*x + c)) - 2*log(sin (d*x + c) + 1)/a^2 + 2*log(sin(d*x + c))/a^2)/d
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} d} - \frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2} d} - \frac {2 \, \sin \left (d x + c\right ) + 1}{{\left (\sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )\right )} a^{2} d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
2*log(abs(sin(d*x + c) + 1))/(a^2*d) - 2*log(abs(sin(d*x + c)))/(a^2*d) - (2*sin(d*x + c) + 1)/((sin(d*x + c)^2 + sin(d*x + c))*a^2*d)
Time = 18.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.00 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^2\,d}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1}{d\,\left (2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d} \] Input:
int(cot(c + d*x)/(sin(c + d*x)*(a + a*sin(c + d*x))^2),x)
Output:
(4*log(tan(c/2 + (d*x)/2) + 1))/(a^2*d) - (2*log(tan(c/2 + (d*x)/2)))/(a^2 *d) - (2*tan(c/2 + (d*x)/2) - 3*tan(c/2 + (d*x)/2)^2 + 1)/(d*(4*a^2*tan(c/ 2 + (d*x)/2)^2 + 2*a^2*tan(c/2 + (d*x)/2)^3 + 2*a^2*tan(c/2 + (d*x)/2))) - tan(c/2 + (d*x)/2)/(2*a^2*d)
Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.82 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+\sin \left (d x +c \right )^{2}-3 \sin \left (d x +c \right )-2}{2 \sin \left (d x +c \right ) a^{2} d \left (\sin \left (d x +c \right )+1\right )} \] Input:
int(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^2,x)
Output:
(8*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))*sin(c + d*x)**2 - 4*log(tan((c + d *x)/2))*sin(c + d*x) + sin(c + d*x)**2 - 3*sin(c + d*x) - 2)/(2*sin(c + d* x)*a**2*d*(sin(c + d*x) + 1))