Integrand size = 13, antiderivative size = 168 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=-\frac {2 b^3 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2}}-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (x))}{2 a^4}+\frac {b \left (2 a^2-3 b^2\right ) \cot (x)}{a^3 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))} \]
-2*b^3*(4*a^2-3*b^2)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/a^4/(a^2-b^2 )^(3/2)-1/2*(a^2+6*b^2)*arctanh(cos(x))/a^4+b*(2*a^2-3*b^2)*cot(x)/a^3/(a^ 2-b^2)-1/2*(a^2-3*b^2)*cot(x)*csc(x)/a^2/(a^2-b^2)-b^2*cot(x)*csc(x)/a/(a^ 2-b^2)/(a+b*sin(x))
Time = 0.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\frac {\frac {16 b^3 \left (-4 a^2+3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+8 a b \cot \left (\frac {x}{2}\right )-a^2 \csc ^2\left (\frac {x}{2}\right )-4 \left (a^2+6 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \left (a^2+6 b^2\right ) \log \left (\sin \left (\frac {x}{2}\right )\right )+a^2 \sec ^2\left (\frac {x}{2}\right )-\frac {8 a b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}-8 a b \tan \left (\frac {x}{2}\right )}{8 a^4} \]
((16*b^3*(-4*a^2 + 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 8*a*b*Cot[x/2] - a^2*Csc[x/2]^2 - 4*(a^2 + 6*b^2)*Log[Cos[x/ 2]] + 4*(a^2 + 6*b^2)*Log[Sin[x/2]] + a^2*Sec[x/2]^2 - (8*a*b^4*Cos[x])/(( a - b)*(a + b)*(a + b*Sin[x])) - 8*a*b*Tan[x/2])/(8*a^4)
Time = 1.18 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 3281, 3042, 3534, 25, 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.
| \(\displaystyle \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^3 (a+b \sin (x))^2}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int \frac {\csc ^3(x) \left (a^2-b \sin (x) a-3 b^2+2 b^2 \sin ^2(x)\right )}{a+b \sin (x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^2-b \sin (x) a-3 b^2+2 b^2 \sin (x)^2}{\sin (x)^3 (a+b \sin (x))}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^2(x) \left (-b \left (a^2-3 b^2\right ) \sin ^2(x)-a \left (a^2+b^2\right ) \sin (x)+2 b \left (2 a^2-3 b^2\right )\right )}{a+b \sin (x)}dx}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^2(x) \left (-b \left (a^2-3 b^2\right ) \sin ^2(x)-a \left (a^2+b^2\right ) \sin (x)+2 b \left (2 a^2-3 b^2\right )\right )}{a+b \sin (x)}dx}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-b \left (a^2-3 b^2\right ) \sin (x)^2-a \left (a^2+b^2\right ) \sin (x)+2 b \left (2 a^2-3 b^2\right )}{\sin (x)^2 (a+b \sin (x))}dx}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {\csc (x) \left (a^4+5 b^2 a^2+b \left (a^2-3 b^2\right ) \sin (x) a-6 b^4\right )}{a+b \sin (x)}dx}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\csc (x) \left (a^4+5 b^2 a^2+b \left (a^2-3 b^2\right ) \sin (x) a-6 b^4\right )}{a+b \sin (x)}dx}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {a^4+5 b^2 a^2+b \left (a^2-3 b^2\right ) \sin (x) a-6 b^4}{\sin (x) (a+b \sin (x))}dx}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\left (a^4+5 a^2 b^2-6 b^4\right ) \int \csc (x)dx}{a}-2 b^3 \left (4 a-\frac {3 b^2}{a}\right ) \int \frac {1}{a+b \sin (x)}dx}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\left (a^4+5 a^2 b^2-6 b^4\right ) \int \csc (x)dx}{a}-2 b^3 \left (4 a-\frac {3 b^2}{a}\right ) \int \frac {1}{a+b \sin (x)}dx}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\left (a^4+5 a^2 b^2-6 b^4\right ) \int \csc (x)dx}{a}-4 b^3 \left (4 a-\frac {3 b^2}{a}\right ) \int \frac {1}{a \tan ^2\left (\frac {x}{2}\right )+2 b \tan \left (\frac {x}{2}\right )+a}d\tan \left (\frac {x}{2}\right )}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {-\frac {8 b^3 \left (4 a-\frac {3 b^2}{a}\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )+\frac {\left (a^4+5 a^2 b^2-6 b^4\right ) \int \csc (x)dx}{a}}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\left (a^4+5 a^2 b^2-6 b^4\right ) \int \csc (x)dx}{a}-\frac {4 b^3 \left (4 a-\frac {3 b^2}{a}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}}{a}-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}}{2 a}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 a}-\frac {-\frac {2 b \left (2 a^2-3 b^2\right ) \cot (x)}{a}-\frac {-\frac {4 b^3 \left (4 a-\frac {3 b^2}{a}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {\left (a^4+5 a^2 b^2-6 b^4\right ) \text {arctanh}(\cos (x))}{a}}{a}}{2 a}}{a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\) |
(-1/2*(-(((-4*b^3*(4*a - (3*b^2)/a)*ArcTan[(2*b + 2*a*Tan[x/2])/(2*Sqrt[a^ 2 - b^2])])/Sqrt[a^2 - b^2] - ((a^4 + 5*a^2*b^2 - 6*b^4)*ArcTanh[Cos[x]])/ a)/a) - (2*b*(2*a^2 - 3*b^2)*Cot[x])/a)/a - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/ (2*a))/(a*(a^2 - b^2)) - (b^2*Cot[x]*Csc[x])/(a*(a^2 - b^2)*(a + b*Sin[x]) )
3.2.92.3.1 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 = 0.84 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(\frac {\frac {a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-4 b \tan \left (\frac {x}{2}\right )}{4 a^{3}}-\frac {4 b^{3} \left (\frac {\frac {b^{2} \tan \left (\frac {x}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a}+\frac {\left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (2 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {x}{2}\right )}\) | \(181\) |
| risch | \(\frac {2 a^{4} {\mathrm e}^{4 i x}+2 a^{2} b^{2} {\mathrm e}^{4 i x}-i a^{3} b \,{\mathrm e}^{5 i x}+3 i a \,b^{3} {\mathrm e}^{5 i x}+8 i a^{3} b \,{\mathrm e}^{3 i x}-12 i a \,b^{3} {\mathrm e}^{3 i x}+2 a^{4} {\mathrm e}^{2 i x}-10 a^{2} b^{2} {\mathrm e}^{2 i x}-7 i a^{3} b \,{\mathrm e}^{i x}+9 i a \,b^{3} {\mathrm e}^{i x}-6 b^{4} {\mathrm e}^{4 i x}+12 b^{4} {\mathrm e}^{2 i x}+4 a^{2} b^{2}-6 b^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left (a^{2}-b^{2}\right ) \left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right ) a^{3}}-\frac {4 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{2}}+\frac {3 i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{4}}+\frac {4 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{2}}-\frac {3 i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{4}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{a^{4}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{a^{4}}\) | \(572\) |
1/4/a^3*(1/2*a*tan(1/2*x)^2-4*b*tan(1/2*x))-4*b^3/a^4*((1/2*b^2/(a^2-b^2)* tan(1/2*x)+1/2*a*b/(a^2-b^2))/(a*tan(1/2*x)^2+2*b*tan(1/2*x)+a)+1/2*(4*a^2 -3*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)))- 1/8/a^2/tan(1/2*x)^2+1/4/a^4*(2*a^2+12*b^2)*ln(tan(1/2*x))+b/a^3/tan(1/2*x )
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (158) = 316\).
Time = 0.74 (sec) , antiderivative size = 1174, normalized size of antiderivative = 6.99 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\text {Too large to display} \]
[-1/4*(4*(2*a^5*b^2 - 5*a^3*b^4 + 3*a*b^6)*cos(x)^3 - 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(x)*sin(x) + 2*(4*a^3*b^3 - 3*a*b^5 - (4*a^3*b^3 - 3*a*b^5) *cos(x)^2 + (4*a^2*b^4 - 3*b^6 - (4*a^2*b^4 - 3*b^6)*cos(x)^2)*sin(x))*sqr t(-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^7 - 6*a^5*b^2 + 11*a^3*b^4 - 6*a*b^6)*cos(x) + (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a* b^6)*cos(x)^2 + (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7 - (a^6*b + 4*a^4*b ^3 - 11*a^2*b^5 + 6*b^7)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 )*cos(x)^2 + (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7 - (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^9 - 2*a ^7*b^2 + a^5*b^4 - (a^9 - 2*a^7*b^2 + a^5*b^4)*cos(x)^2 + (a^8*b - 2*a^6*b ^3 + a^4*b^5 - (a^8*b - 2*a^6*b^3 + a^4*b^5)*cos(x)^2)*sin(x)), -1/4*(4*(2 *a^5*b^2 - 5*a^3*b^4 + 3*a*b^6)*cos(x)^3 - 6*(a^6*b - 2*a^4*b^3 + a^2*b^5) *cos(x)*sin(x) - 4*(4*a^3*b^3 - 3*a*b^5 - (4*a^3*b^3 - 3*a*b^5)*cos(x)^2 + (4*a^2*b^4 - 3*b^6 - (4*a^2*b^4 - 3*b^6)*cos(x)^2)*sin(x))*sqrt(a^2 - b^2 )*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + 2*(a^7 - 6*a^5*b^2 + 11*a^3*b^4 - 6*a*b^6)*cos(x) + (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - ( a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(x)^2 + (a^6*b + 4*a^4*b^3 -...
\[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\text {Exception raised: ValueError} \]
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 = 215, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=-\frac {2 \, {\left (4 \, a^{2} b^{3} - 3 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (b^{5} \tan \left (\frac {1}{2} \, x\right ) + a b^{4}\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}} + \frac {{\left (a^{2} + 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 36 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}}{8 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
-2*(4*a^2*b^3 - 3*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^6 - a^4*b^2)*sqrt(a^2 - b^2)) - 2*(b^5*tan (1/2*x) + a*b^4)/((a^6 - a^4*b^2)*(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)) + 1/2*(a^2 + 6*b^2)*log(abs(tan(1/2*x)))/a^4 + 1/8*(a^2*tan(1/2*x)^2 - 8*a* b*tan(1/2*x))/a^4 - 1/8*(6*a^2*tan(1/2*x)^2 + 36*b^2*tan(1/2*x)^2 - 8*a*b* tan(1/2*x) + a^2)/(a^4*tan(1/2*x)^2)
Time = 7.64 (sec) , antiderivative size = 1576, normalized size of antiderivative = 9.38 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^2} \, dx=\text {Too large to display} \]
tan(x/2)^2/(8*a^2) - (a^2/2 - 3*a*b*tan(x/2) + (tan(x/2)^2*(a^4 + 32*b^4 - 17*a^2*b^2))/(2*(a^2 - b^2)) + (4*b*tan(x/2)^3*(2*b^4 - a^4 + a^2*b^2))/( a*(a^2 - b^2)))/(4*a^4*tan(x/2)^2 + 4*a^4*tan(x/2)^4 + 8*a^3*b*tan(x/2)^3) + (log(tan(x/2))*(a^2 + 6*b^2))/(2*a^4) - (b*tan(x/2))/a^3 + (b^3*atan((( b^3*(4*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((a^8*b - 12*a^4*b^5 + 13 *a^6*b^3)/(a^8 - a^6*b^2) - (tan(x/2)*(a^10 - 24*a^2*b^8 + 56*a^4*b^6 - 35 *a^6*b^4 + 2*a^8*b^2))/(a^9 + a^5*b^4 - 2*a^7*b^2) + (b^3*(4*a^2 - 3*b^2)* (-(a + b)^3*(a - b)^3)^(1/2)*((2*a^10*b - 2*a^8*b^3)/(a^8 - a^6*b^2) - (ta n(x/2)*(6*a^12 - 8*a^6*b^6 + 22*a^8*b^4 - 20*a^10*b^2))/(a^9 + a^5*b^4 - 2 *a^7*b^2)))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2) - (b^3*(4*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/ 2)*((tan(x/2)*(a^10 - 24*a^2*b^8 + 56*a^4*b^6 - 35*a^6*b^4 + 2*a^8*b^2))/( a^9 + a^5*b^4 - 2*a^7*b^2) - (a^8*b - 12*a^4*b^5 + 13*a^6*b^3)/(a^8 - a^6* b^2) + (b^3*(4*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((2*a^10*b - 2*a^ 8*b^3)/(a^8 - a^6*b^2) - (tan(x/2)*(6*a^12 - 8*a^6*b^6 + 22*a^8*b^4 - 20*a ^10*b^2))/(a^9 + a^5*b^4 - 2*a^7*b^2)))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^ 8*b^2))*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))/((2*(21*a^2*b^5 - 18 *b^7 + 4*a^4*b^3))/(a^8 - a^6*b^2) + (2*tan(x/2)*(18*b^8 - 30*a^2*b^6 + 8* a^4*b^4))/(a^9 + a^5*b^4 - 2*a^7*b^2) + (b^3*(4*a^2 - 3*b^2)*(-(a + b)^3*( a - b)^3)^(1/2)*((a^8*b - 12*a^4*b^5 + 13*a^6*b^3)/(a^8 - a^6*b^2) - (t...