Integrand size = 11, antiderivative size = 38 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^2}+\frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2} \] Output:
-arctanh(cos(x))/a^2+4/3*cos(x)/a^2/(1+sin(x))+1/3*cos(x)/(a+a*sin(x))^2
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {-3 \text {arctanh}(\cos (x))+\frac {\cos (x) (5+4 \sin (x))}{(1+\sin (x))^2}}{3 a^2} \] Input:
Integrate[Csc[x]/(a + a*Sin[x])^2,x]
Output:
(-3*ArcTanh[Cos[x]] + (Cos[x]*(5 + 4*Sin[x]))/(1 + Sin[x])^2)/(3*a^2)
Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3245, 3042, 3457, 27, 3042, 4257}
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.
\(\displaystyle \int \frac {\csc (x)}{(a \sin (x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (x) (a \sin (x)+a)^2}dx\) |
\(\Big \downarrow \) 3245 |
\(\displaystyle \frac {\int \frac {\csc (x) (3 a-a \sin (x))}{\sin (x) a+a}dx}{3 a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {3 a-a \sin (x)}{\sin (x) (\sin (x) a+a)}dx}{3 a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle \frac {\frac {\int 3 a^2 \csc (x)dx}{a^2}+\frac {4 \cos (x)}{\sin (x)+1}}{3 a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \int \csc (x)dx+\frac {4 \cos (x)}{\sin (x)+1}}{3 a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \int \csc (x)dx+\frac {4 \cos (x)}{\sin (x)+1}}{3 a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\frac {4 \cos (x)}{\sin (x)+1}-3 \text {arctanh}(\cos (x))}{3 a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2}\) |
Input:
Int[Csc[x]/(a + a*Sin[x])^2,x]
Output:
Cos[x]/(3*(a + a*Sin[x])^2) + (-3*ArcTanh[Cos[x]] + (4*Cos[x])/(1 + Sin[x] ))/(3*a^2)
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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\tan \left (\frac {x}{2}\right )+1}+\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(41\) |
parallelrisch | \(\frac {3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {x}{2}\right )\right )+12 \tan \left (\frac {x}{2}\right )^{2}+18 \tan \left (\frac {x}{2}\right )+10}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(45\) |
norman | \(\frac {\frac {4 \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {10}{3 a}+\frac {6 \tan \left (\frac {x}{2}\right )}{a}}{a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(49\) |
risch | \(\frac {6 i {\mathrm e}^{i x}+2 \,{\mathrm e}^{2 i x}-\frac {8}{3}}{\left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{2}}\) | \(59\) |
Input:
int(csc(x)/(a+a*sin(x))^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*(4/3/(tan(1/2*x)+1)^3-2/(tan(1/2*x)+1)^2+4/(tan(1/2*x)+1)+ln(tan(1/2 *x)))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.08 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=-\frac {8 \, \cos \left (x\right )^{2} + 3 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (4 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + 10 \, \cos \left (x\right ) + 2}{6 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)/(a+a*sin(x))^2,x, algorithm="fricas")
Output:
-1/6*(8*cos(x)^2 + 3*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*log(1/2 *cos(x) + 1/2) - 3*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*log(-1/2* cos(x) + 1/2) + 2*(4*cos(x) - 1)*sin(x) + 10*cos(x) + 2)/(a^2*cos(x)^2 - a ^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))
\[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\int \frac {\csc {\left (x \right )}}{\sin ^{2}{\left (x \right )} + 2 \sin {\left (x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(csc(x)/(a+a*sin(x))**2,x)
Output:
Integral(csc(x)/(sin(x)**2 + 2*sin(x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (34) = 68\).
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {2 \, {\left (\frac {9 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 5\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \] Input:
integrate(csc(x)/(a+a*sin(x))^2,x, algorithm="maxima")
Output:
2/3*(9*sin(x)/(cos(x) + 1) + 6*sin(x)^2/(cos(x) + 1)^2 + 5)/(a^2 + 3*a^2*s in(x)/(cos(x) + 1) + 3*a^2*sin(x)^2/(cos(x) + 1)^2 + a^2*sin(x)^3/(cos(x) + 1)^3) + log(sin(x)/(cos(x) + 1))/a^2
Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) + 5\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \] Input:
integrate(csc(x)/(a+a*sin(x))^2,x, algorithm="giac")
Output:
log(abs(tan(1/2*x)))/a^2 + 2/3*(6*tan(1/2*x)^2 + 9*tan(1/2*x) + 5)/(a^2*(t an(1/2*x) + 1)^3)
Time = 17.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {10}{3}}{a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \] Input:
int(1/(sin(x)*(a + a*sin(x))^2),x)
Output:
log(tan(x/2))/a^2 + (6*tan(x/2) + 4*tan(x/2)^2 + 10/3)/(a^2*(tan(x/2) + 1) ^3)
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {3 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )^{3}+9 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )^{2}+9 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )+3 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right )-4 \tan \left (\frac {x}{2}\right )^{3}+6 \tan \left (\frac {x}{2}\right )+6}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )^{3}+3 \tan \left (\frac {x}{2}\right )^{2}+3 \tan \left (\frac {x}{2}\right )+1\right )} \] Input:
int(csc(x)/(a+a*sin(x))^2,x)
Output:
(3*log(tan(x/2))*tan(x/2)**3 + 9*log(tan(x/2))*tan(x/2)**2 + 9*log(tan(x/2 ))*tan(x/2) + 3*log(tan(x/2)) - 4*tan(x/2)**3 + 6*tan(x/2) + 6)/(3*a**2*(t an(x/2)**3 + 3*tan(x/2)**2 + 3*tan(x/2) + 1))