Integrand size = 11, antiderivative size = 145 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}}-\frac {\text {arctanh}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
-b*(6*a^4-5*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/a^3/(a ^2-b^2)^(5/2)-arctanh(cos(x))/a^3-1/2*b^2*cos(x)/a/(a^2-b^2)/(a+b*sin(x))^ 2-1/2*b^2*(5*a^2-2*b^2)*cos(x)/a^2/(a^2-b^2)^2/(a+b*sin(x))
Time = 0.70 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+2 \log \left (\cos \left (\frac {x}{2}\right )\right )-2 \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {a b^2 \cos (x) \left (6 a^3-3 a b^2+b \left (5 a^2-2 b^2\right ) \sin (x)\right )}{(a-b)^2 (a+b)^2 (a+b \sin (x))^2}}{2 a^3} \]
-1/2*((2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 2*Log[Cos[x/2]] - 2*Log[Sin[x/2]] + (a*b^2*Cos[ x]*(6*a^3 - 3*a*b^2 + b*(5*a^2 - 2*b^2)*Sin[x]))/((a - b)^2*(a + b)^2*(a + b*Sin[x])^2))/a^3
Time = 0.90 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.34, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3281, 3042, 3534, 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.
| \(\displaystyle \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x) (a+b \sin (x))^3}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int \frac {\csc (x) \left (b^2 \sin ^2(x)-2 a b \sin (x)+2 \left (a^2-b^2\right )\right )}{(a+b \sin (x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b^2 \sin (x)^2-2 a b \sin (x)+2 \left (a^2-b^2\right )}{\sin (x) (a+b \sin (x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\csc (x) \left (2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \sin (x)\right )}{a+b \sin (x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \sin (x)}{\sin (x) (a+b \sin (x))}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2-b^2\right )^2 \int \csc (x)dx}{a}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2-b^2\right )^2 \int \csc (x)dx}{a}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \sin (x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2-b^2\right )^2 \int \csc (x)dx}{a}-\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {x}{2}\right )+2 b \tan \left (\frac {x}{2}\right )+a}d\tan \left (\frac {x}{2}\right )}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2-b^2\right )^2 \int \csc (x)dx}{a}+\frac {4 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {2 \left (a^2-b^2\right )^2 \int \csc (x)dx}{a}-\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (a^2-b^2\right )^2 \text {arctanh}(\cos (x))}{a}-\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
-1/2*(b^2*Cos[x])/(a*(a^2 - b^2)*(a + b*Sin[x])^2) + (((-2*b*(6*a^4 - 5*a^ 2*b^2 + 2*b^4)*ArcTan[(2*b + 2*a*Tan[x/2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a ^2 - b^2]) - (2*(a^2 - b^2)^2*ArcTanh[Cos[x]])/a)/(a*(a^2 - b^2)) - (b^2*( 5*a^2 - 2*b^2)*Cos[x])/(a*(a^2 - b^2)*(a + b*Sin[x])))/(2*a*(a^2 - b^2))
3.2.99.3.1 Defintions of rubi rules used
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 = 0.93 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.86
| method | result | size |
| default | \(-\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}+\frac {3 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {a \,b^{2} \left (17 a^{2}-8 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}+\frac {3 a^{2} b \left (2 a^{2}-b^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{a^{3}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(270\) |
| risch | \(\frac {i b \left (-4 i a^{3} b \,{\mathrm e}^{3 i x}+i a \,b^{3} {\mathrm e}^{3 i x}+16 i a^{3} b \,{\mathrm e}^{i x}-7 i a \,b^{3} {\mathrm e}^{i x}+10 a^{4} {\mathrm e}^{2 i x}+a^{2} b^{2} {\mathrm e}^{2 i x}-2 b^{4} {\mathrm e}^{2 i x}-5 a^{2} b^{2}+2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} a^{2} \left (a^{2}-b^{2}\right )^{2}}+\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}+\frac {i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}-\frac {i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}\) | \(623\) |
-2*b/a^3*((1/2*a*b^2*(7*a^2-4*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3+3/2*b* (2*a^4+3*a^2*b^2-2*b^4)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2+1/2*a*b^2*(17*a^2 -8*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)+3/2*a^2*b*(2*a^2-b^2)/(a^4-2*a^2*b^ 2+b^4))/(a*tan(1/2*x)^2+2*b*tan(1/2*x)+a)^2+1/2*(6*a^4-5*a^2*b^2+2*b^4)/(a ^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*x)+2*b)/(a^2-b^2 )^(1/2)))+1/a^3*ln(tan(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (135) = 270\).
Time = 0.83 (sec) , antiderivative size = 1027, normalized size of antiderivative = 7.08 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]
[-1/4*(2*(5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(x)*sin(x) + (6*a^6*b + a^4* b^3 - 3*a^2*b^5 + 2*b^7 - (6*a^4*b^3 - 5*a^2*b^5 + 2*b^7)*cos(x)^2 + 2*(6* a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*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))*sqr t(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 6*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(x) + 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6 *b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^ 3*b^5 - a*b^7)*sin(x))*log(1/2*cos(x) + 1/2) - 2*(a^8 - 2*a^6*b^2 + 2*a^2* b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^11 - 2*a ^9*b^2 + 2*a^5*b^6 - a^3*b^8 - (a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8) *cos(x)^2 + 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*sin(x)), -1/2*((5 *a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(x)*sin(x) - (6*a^6*b + a^4*b^3 - 3*a^2 *b^5 + 2*b^7 - (6*a^4*b^3 - 5*a^2*b^5 + 2*b^7)*cos(x)^2 + 2*(6*a^5*b^2 - 5 *a^3*b^4 + 2*a*b^6)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a ^2 - b^2)*cos(x))) + 3*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(x) + (a^8 - 2 *a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x )^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*log(1/2*cos(x) + 1 /2) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^ 6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*l...
\[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\int \frac {\csc {\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\text {Exception raised: ValueError} \]
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 = 0.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.70 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=-\frac {{\left (6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {7 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, a b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 6 \, b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 17 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 8 \, a b^{5} \tan \left (\frac {1}{2} \, x\right ) + 6 \, a^{4} b^{2} - 3 \, a^{2} b^{4}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} \]
-(6*a^4*b - 5*a^2*b^3 + 2*b^5)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan(( a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(a^2 - b^2)) - (7*a^3*b^3*tan(1/2*x)^3 - 4*a*b^5*tan(1/2*x)^3 + 6*a^4*b^2*tan(1 /2*x)^2 + 9*a^2*b^4*tan(1/2*x)^2 - 6*b^6*tan(1/2*x)^2 + 17*a^3*b^3*tan(1/2 *x) - 8*a*b^5*tan(1/2*x) + 6*a^4*b^2 - 3*a^2*b^4)/((a^7 - 2*a^5*b^2 + a^3* b^4)*(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)^2) + log(abs(tan(1/2*x)))/a^3
Time = 11.48 (sec) , antiderivative size = 2191, normalized size of antiderivative = 15.11 \[ \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]
((3*(b^4 - 2*a^2*b^2))/(a*(a^4 + b^4 - 2*a^2*b^2)) - (3*tan(x/2)^2*(3*a^2* b^4 - 2*b^6 + 2*a^4*b^2))/(a^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(x/2)*(8*b^5 - 17*a^2*b^3))/(a^2*(a^4 + b^4 - 2*a^2*b^2)) + (b*tan(x/2)^3*(4*b^4 - 7*a ^2*b^2))/(a^2*(a^4 + b^4 - 2*a^2*b^2)))/(tan(x/2)^2*(2*a^2 + 4*b^2) + a^2 + a^2*tan(x/2)^4 + 4*a*b*tan(x/2) + 4*a*b*tan(x/2)^3) + log(tan(x/2))/a^3 + (b*atan(((b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*a^7*b + 4*a^3*b^5 - 9*a^5*b ^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) + (tan(x/2)*(8*a*b^10 - 2*a^11 - 36*a^3*b^ 8 + 68*a^5*b^6 - 62*a^7*b^4 + 24*a^9*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6 *a^7*b^4 - 4*a^9*b^2) - (b*((2*a^10*b + 2*a^6*b^5 - 4*a^8*b^3)/(a^8 + a^4* b^4 - 2*a^6*b^2) - (tan(x/2)*(6*a^14 - 8*a^4*b^10 + 38*a^6*b^8 - 72*a^8*b^ 6 + 68*a^10*b^4 - 32*a^12*b^2))/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4 + 2*b^4 - 5*a^2*b^2))/(2*( a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(6*a ^4 + 2*b^4 - 5*a^2*b^2)*1i)/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*a^7*b + 4 *a^3*b^5 - 9*a^5*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) + (tan(x/2)*(8*a*b^10 - 2*a^11 - 36*a^3*b^8 + 68*a^5*b^6 - 62*a^7*b^4 + 24*a^9*b^2))/(a^11 + a^3*b ^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) + (b*((2*a^10*b + 2*a^6*b^5 - 4*a^ 8*b^3)/(a^8 + a^4*b^4 - 2*a^6*b^2) - (tan(x/2)*(6*a^14 - 8*a^4*b^10 + 38*a ^6*b^8 - 72*a^8*b^6 + 68*a^10*b^4 - 32*a^12*b^2))/(a^11 + a^3*b^8 - 4*a...