Integrand size = 13, antiderivative size = 123 \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\frac {2 b^2 \left (3 a^2-2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2}}+\frac {2 b \text {arctanh}(\cos (x))}{a^3}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))} \] Output:
2*b^2*(3*a^2-2*b^2)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/a^3/(a^2-b^2) ^(3/2)+2*b*arctanh(cos(x))/a^3-(a^2-2*b^2)*cot(x)/a^2/(a^2-b^2)-b^2*cot(x) /a/(a^2-b^2)/(a+b*sin(x))
Time = 1.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\frac {\frac {4 b^2 \left (3 a^2-2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-a \cot \left (\frac {x}{2}\right )+4 b \log \left (\cos \left (\frac {x}{2}\right )\right )-4 b \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {2 a b^3 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}+a \tan \left (\frac {x}{2}\right )}{2 a^3} \] Input:
Integrate[Csc[x]^2/(a + b*Sin[x])^2,x]
Output:
((4*b^2*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b ^2)^(3/2) - a*Cot[x/2] + 4*b*Log[Cos[x/2]] - 4*b*Log[Sin[x/2]] + (2*a*b^3* Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])) + a*Tan[x/2])/(2*a^3)
Time = 0.89 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.21, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 3281, 3042, 3534, 25, 3042, 3480, 3042, 3139, 1083, 217, 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.
| \(\displaystyle \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^2 (a+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int \frac {\csc ^2(x) \left (a^2-b \sin (x) a-2 b^2+b^2 \sin ^2(x)\right )}{a+b \sin (x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^2-b \sin (x) a-2 b^2+b^2 \sin (x)^2}{\sin (x)^2 (a+b \sin (x))}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc (x) \left (2 b \left (a^2-b^2\right )-a b^2 \sin (x)\right )}{a+b \sin (x)}dx}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc (x) \left (2 b \left (a^2-b^2\right )-a b^2 \sin (x)\right )}{a+b \sin (x)}dx}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {2 b \left (a^2-b^2\right )-a b^2 \sin (x)}{\sin (x) (a+b \sin (x))}dx}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {\frac {2 b \left (a^2-b^2\right ) \int \csc (x)dx}{a}-\frac {b^2 \left (3 a^2-2 b^2\right ) \int \frac {1}{a+b \sin (x)}dx}{a}}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {2 b \left (a^2-b^2\right ) \int \csc (x)dx}{a}-\frac {b^2 \left (3 a^2-2 b^2\right ) \int \frac {1}{a+b \sin (x)}dx}{a}}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {\frac {2 b \left (a^2-b^2\right ) \int \csc (x)dx}{a}-\frac {2 b^2 \left (3 a^2-2 b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {x}{2}\right )+2 b \tan \left (\frac {x}{2}\right )+a}d\tan \left (\frac {x}{2}\right )}{a}}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 \left (3 a^2-2 b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{a}+\frac {2 b \left (a^2-b^2\right ) \int \csc (x)dx}{a}}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {2 b \left (a^2-b^2\right ) \int \csc (x)dx}{a}-\frac {2 b^2 \left (3 a^2-2 b^2\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {-\frac {2 b^2 \left (3 a^2-2 b^2\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {2 b \left (a^2-b^2\right ) \text {arctanh}(\cos (x))}{a}}{a}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
Input:
Int[Csc[x]^2/(a + b*Sin[x])^2,x]
Output:
(-(((-2*b^2*(3*a^2 - 2*b^2)*ArcTan[(2*b + 2*a*Tan[x/2])/(2*Sqrt[a^2 - b^2] )])/(a*Sqrt[a^2 - b^2]) - (2*b*(a^2 - b^2)*ArcTanh[Cos[x]])/a)/a) - ((a^2 - 2*b^2)*Cot[x])/a)/(a*(a^2 - b^2)) - (b^2*Cot[x])/(a*(a^2 - b^2)*(a + b*S in[x]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
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 + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 ))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2* n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(\frac {\tan \left (\frac {x}{2}\right )}{2 a^{2}}-\frac {1}{2 a^{2} \tan \left (\frac {x}{2}\right )}-\frac {2 b \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}+\frac {2 b^{2} \left (\frac {\frac {b^{2} \tan \left (\frac {x}{2}\right )}{a^{2}-b^{2}}+\frac {a b}{a^{2}-b^{2}}}{a \tan \left (\frac {x}{2}\right )^{2}+2 b \tan \left (\frac {x}{2}\right )+a}+\frac {\left (3 a^{2}-2 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{3}}\) | \(144\) |
| risch | \(-\frac {2 i \left (2 a^{3} {\mathrm e}^{i x}-3 b^{2} a \,{\mathrm e}^{i x}-i a^{2} b \,{\mathrm e}^{2 i x}+2 i b^{3} {\mathrm e}^{2 i x}+i a^{2} b -2 i b^{3}+a \,b^{2} {\mathrm e}^{3 i x}\right )}{\left ({\mathrm e}^{2 i x}-1\right ) \left (a^{2}-b^{2}\right ) \left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right ) a^{2}}-\frac {2 b \ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}+\frac {2 b \ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) a}-\frac {2 b^{4} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{3}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) a}+\frac {2 b^{4} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{3}}\) | \(456\) |
Input:
int(csc(x)^2/(a+b*sin(x))^2,x,method=_RETURNVERBOSE)
Output:
1/2/a^2*tan(1/2*x)-1/2/a^2/tan(1/2*x)-2/a^3*b*ln(tan(1/2*x))+2/a^3*b^2*((b ^2/(a^2-b^2)*tan(1/2*x)+a*b/(a^2-b^2))/(a*tan(1/2*x)^2+2*b*tan(1/2*x)+a)+( 3*a^2-2*b^2)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2)^(1/ 2)))
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (117) = 234\).
Time = 0.27 (sec) , antiderivative size = 784, normalized size of antiderivative = 6.37 \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx =\text {Too large to display} \] Input:
integrate(csc(x)^2/(a+b*sin(x))^2,x, algorithm="fricas")
Output:
[-1/2*(2*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(x)*sin(x) + (3*a^2*b^3 - 2*b^5 - (3*a^2*b^3 - 2*b^5)*cos(x)^2 + (3*a^3*b^2 - 2*a*b^4)*sin(x))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x) *sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*cos(x) - 2*(a^4*b^2 - 2*a^2*b^4 + b ^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*si n(x))*log(1/2*cos(x) + 1/2) + 2*(a^4*b^2 - 2*a^2*b^4 + b^6 - (a^4*b^2 - 2* a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(-1/2*cos (x) + 1/2))/(a^7*b - 2*a^5*b^3 + a^3*b^5 - (a^7*b - 2*a^5*b^3 + a^3*b^5)*c os(x)^2 + (a^8 - 2*a^6*b^2 + a^4*b^4)*sin(x)), -((a^5*b - 3*a^3*b^3 + 2*a* b^5)*cos(x)*sin(x) + (3*a^2*b^3 - 2*b^5 - (3*a^2*b^3 - 2*b^5)*cos(x)^2 + ( 3*a^3*b^2 - 2*a*b^4)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt( a^2 - b^2)*cos(x))) + (a^6 - 2*a^4*b^2 + a^2*b^4)*cos(x) - (a^4*b^2 - 2*a^ 2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(1/2*cos(x) + 1/2) + (a^4*b^2 - 2*a^2*b^4 + b^6 - (a^4*b ^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(- 1/2*cos(x) + 1/2))/(a^7*b - 2*a^5*b^3 + a^3*b^5 - (a^7*b - 2*a^5*b^3 + a^3 *b^5)*cos(x)^2 + (a^8 - 2*a^6*b^2 + a^4*b^4)*sin(x))]
\[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\int \frac {\csc ^{2}{\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(csc(x)**2/(a+b*sin(x))**2,x)
Output:
Integral(csc(x)**2/(a + b*sin(x))**2, x)
Exception generated. \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(csc(x)^2/(a+b*sin(x))^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.14 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.90 \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\frac {2 \, {\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, a^{3} b \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, a b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 3 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 11 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, a^{3} b \tan \left (\frac {1}{2} \, x\right ) + 14 \, a b^{3} \tan \left (\frac {1}{2} \, x\right ) - 3 \, a^{4} + 3 \, a^{2} b^{2}}{6 \, {\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x\right )^{2} + a \tan \left (\frac {1}{2} \, x\right )\right )}} - \frac {2 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a^{2}} \] Input:
integrate(csc(x)^2/(a+b*sin(x))^2,x, algorithm="giac")
Output:
2*(3*a^2*b^2 - 2*b^4)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2 *x) + b)/sqrt(a^2 - b^2)))/((a^5 - a^3*b^2)*sqrt(a^2 - b^2)) + 1/6*(4*a^3* b*tan(1/2*x)^3 - 4*a*b^3*tan(1/2*x)^3 - 3*a^4*tan(1/2*x)^2 + 11*a^2*b^2*ta n(1/2*x)^2 + 4*b^4*tan(1/2*x)^2 - 2*a^3*b*tan(1/2*x) + 14*a*b^3*tan(1/2*x) - 3*a^4 + 3*a^2*b^2)/((a^5 - a^3*b^2)*(a*tan(1/2*x)^3 + 2*b*tan(1/2*x)^2 + a*tan(1/2*x))) - 2*b*log(abs(tan(1/2*x)))/a^3 + 1/2*tan(1/2*x)/a^2
Time = 17.56 (sec) , antiderivative size = 1471, normalized size of antiderivative = 11.96 \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\text {Too large to display} \] Input:
int(1/(sin(x)^2*(a + b*sin(x))^2),x)
Output:
tan(x/2)/(2*a^2) - (a - (tan(x/2)^2*(4*b^4 - a^4 + a^2*b^2))/(a*(a^2 - b^2 )) + (2*b*tan(x/2)*(a^2 - 3*b^2))/(a^2 - b^2))/(2*a^3*tan(x/2) + 2*a^3*tan (x/2)^3 + 4*a^2*b*tan(x/2)^2) - (2*b*log(tan(x/2)))/a^3 - (b^2*atan(((b^2* (3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((2*tan(x/2)*(8*a*b^7 - 2*a^7 *b - 20*a^3*b^5 + 14*a^5*b^3))/(a^7 + a^3*b^4 - 2*a^5*b^2) - (2*(4*a^3*b^4 - 5*a^5*b^2))/(a^6 - a^4*b^2) + (b^2*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^ 3)^(1/2)*((2*(a^8*b - a^6*b^3))/(a^6 - a^4*b^2) - (2*tan(x/2)*(3*a^10 - 4* a^4*b^6 + 11*a^6*b^4 - 10*a^8*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2)))/(a^9 - a ^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*1i)/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^ 2) - (b^2*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((2*(4*a^3*b^4 - 5* a^5*b^2))/(a^6 - a^4*b^2) - (2*tan(x/2)*(8*a*b^7 - 2*a^7*b - 20*a^3*b^5 + 14*a^5*b^3))/(a^7 + a^3*b^4 - 2*a^5*b^2) + (b^2*(3*a^2 - 2*b^2)*(-(a + b)^ 3*(a - b)^3)^(1/2)*((2*(a^8*b - a^6*b^3))/(a^6 - a^4*b^2) - (2*tan(x/2)*(3 *a^10 - 4*a^4*b^6 + 11*a^6*b^4 - 10*a^8*b^2))/(a^7 + a^3*b^4 - 2*a^5*b^2)) )/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*1i)/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))/((4*tan(x/2)*(4*b^6 - 6*a^2*b^4))/(a^7 + a^3*b^4 - 2*a^5*b^2 ) - (4*(4*b^5 - 6*a^2*b^3))/(a^6 - a^4*b^2) + (b^2*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((2*tan(x/2)*(8*a*b^7 - 2*a^7*b - 20*a^3*b^5 + 14*a^ 5*b^3))/(a^7 + a^3*b^4 - 2*a^5*b^2) - (2*(4*a^3*b^4 - 5*a^5*b^2))/(a^6 - a ^4*b^2) + (b^2*(3*a^2 - 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((2*(a^8*b ...
Time = 0.17 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.98 \[ \int \frac {\csc ^2(x)}{(a+b \sin (x))^2} \, dx=\frac {6 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (x \right )^{2} a^{2} b^{3}-4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (x \right )^{2} b^{5}+6 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (x \right ) a^{3} b^{2}-4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (x \right ) a \,b^{4}-\cos \left (x \right ) \sin \left (x \right ) a^{5} b +3 \cos \left (x \right ) \sin \left (x \right ) a^{3} b^{3}-2 \cos \left (x \right ) \sin \left (x \right ) a \,b^{5}-\cos \left (x \right ) a^{6}+2 \cos \left (x \right ) a^{4} b^{2}-\cos \left (x \right ) a^{2} b^{4}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} a^{4} b^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} a^{2} b^{4}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} b^{6}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right ) a^{5} b +4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right ) a^{3} b^{3}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right ) a \,b^{5}}{\sin \left (x \right ) a^{3} \left (\sin \left (x \right ) a^{4} b -2 \sin \left (x \right ) a^{2} b^{3}+\sin \left (x \right ) b^{5}+a^{5}-2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:
int(csc(x)^2/(a+b*sin(x))^2,x)
Output:
(6*sqrt(a**2 - b**2)*atan((tan(x/2)*a + b)/sqrt(a**2 - b**2))*sin(x)**2*a* *2*b**3 - 4*sqrt(a**2 - b**2)*atan((tan(x/2)*a + b)/sqrt(a**2 - b**2))*sin (x)**2*b**5 + 6*sqrt(a**2 - b**2)*atan((tan(x/2)*a + b)/sqrt(a**2 - b**2)) *sin(x)*a**3*b**2 - 4*sqrt(a**2 - b**2)*atan((tan(x/2)*a + b)/sqrt(a**2 - b**2))*sin(x)*a*b**4 - cos(x)*sin(x)*a**5*b + 3*cos(x)*sin(x)*a**3*b**3 - 2*cos(x)*sin(x)*a*b**5 - cos(x)*a**6 + 2*cos(x)*a**4*b**2 - cos(x)*a**2*b* *4 - 2*log(tan(x/2))*sin(x)**2*a**4*b**2 + 4*log(tan(x/2))*sin(x)**2*a**2* b**4 - 2*log(tan(x/2))*sin(x)**2*b**6 - 2*log(tan(x/2))*sin(x)*a**5*b + 4* log(tan(x/2))*sin(x)*a**3*b**3 - 2*log(tan(x/2))*sin(x)*a*b**5)/(sin(x)*a* *3*(sin(x)*a**4*b - 2*sin(x)*a**2*b**3 + sin(x)*b**5 + a**5 - 2*a**3*b**2 + a*b**4))