Integrand size = 12, antiderivative size = 170 \[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2} d}-\frac {b^2 \cot (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \csc (c+d x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \csc (c+d x))} \]
x/a^3+b*(6*a^4-5*a^2*b^2+2*b^4)*arctanh((a+b*tan(1/2*d*x+1/2*c))/(a^2-b^2) ^(1/2))/a^3/(a^2-b^2)^(5/2)/d-1/2*b^2*cot(d*x+c)/a/(a^2-b^2)/d/(a+b*csc(d* x+c))^2-1/2*b^2*(5*a^2-2*b^2)*cot(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*csc(d*x+c) )
Time = 1.97 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=\frac {\csc ^2(c+d x) (b+a \sin (c+d x)) \left (\frac {a b^3 \cot (c+d x)}{(a-b) (a+b)}-\frac {3 a b^2 \left (2 a^2-b^2\right ) \cot (c+d x) (b+a \sin (c+d x))}{(a-b)^2 (a+b)^2}+2 (c+d x) \csc (c+d x) (b+a \sin (c+d x))^2-\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {a+b \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right ) \csc (c+d x) (b+a \sin (c+d x))^2}{\left (-a^2+b^2\right )^{5/2}}\right )}{2 a^3 d (a+b \csc (c+d x))^3} \]
(Csc[c + d*x]^2*(b + a*Sin[c + d*x])*((a*b^3*Cot[c + d*x])/((a - b)*(a + b )) - (3*a*b^2*(2*a^2 - b^2)*Cot[c + d*x]*(b + a*Sin[c + d*x]))/((a - b)^2* (a + b)^2) + 2*(c + d*x)*Csc[c + d*x]*(b + a*Sin[c + d*x])^2 - (2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(a + b*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cs c[c + d*x]*(b + a*Sin[c + d*x])^2)/(-a^2 + b^2)^(5/2)))/(2*a^3*d*(a + b*Cs c[c + d*x])^3)
Time = 0.99 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.32, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3042, 4272, 25, 3042, 4548, 25, 3042, 4407, 3042, 4318, 3042, 3139, 1083, 219}
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.
| \(\displaystyle \int \frac {1}{(a+b \csc (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \csc (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4272 |
| \(\displaystyle -\frac {\int -\frac {b^2 \csc ^2(c+d x)-2 a b \csc (c+d x)+2 \left (a^2-b^2\right )}{(a+b \csc (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b^2 \csc ^2(c+d x)-2 a b \csc (c+d x)+2 \left (a^2-b^2\right )}{(a+b \csc (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b^2 \csc (c+d x)^2-2 a b \csc (c+d x)+2 \left (a^2-b^2\right )}{(a+b \csc (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \csc (c+d x)}{a+b \csc (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \csc (c+d x)}{a+b \csc (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \csc (c+d x)}{a+b \csc (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (a^2-b^2\right )^2}{a}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {\csc (c+d x)}{a+b \csc (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (a^2-b^2\right )^2}{a}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {\csc (c+d x)}{a+b \csc (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (a^2-b^2\right )^2}{a}-\frac {\left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{\frac {a \sin (c+d x)}{b}+1}dx}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (a^2-b^2\right )^2}{a}-\frac {\left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{\frac {a \sin (c+d x)}{b}+1}dx}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (a^2-b^2\right )^2}{a}-\frac {2 \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )}{b}+1}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {4 \left (6 a^4-5 a^2 b^2+2 b^4\right ) \int \frac {1}{-\left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (1-\frac {a^2}{b^2}\right )}d\left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {2 x \left (a^2-b^2\right )^2}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (a^2-b^2\right )^2}{a}+\frac {2 b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cot (c+d x)}{a d \left (a^2-b^2\right ) (a+b \csc (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \csc (c+d x))^2}\) |
-1/2*(b^2*Cot[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^2) + (((2*(a ^2 - b^2)^2*x)/a + (2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(b*((2*a)/b + 2*Tan[(c + d*x)/2]))/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d))/(a*(a^2 - b^2)) - (b^2*(5*a^2 - 2*b^2)*Cot[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])))/(2*a*(a^2 - b^2))
3.1.50.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[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[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(n + 1)*(a^2 - b^2)) Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x ], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ erQ[2*n]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^( m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Time = 1.05 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.85
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {\frac {4 a^{2} b \left (4 a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {4 a \left (10 a^{4}+a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {4 a^{2} b \left (16 a^{2}-7 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {4 a \,b^{2} \left (5 a^{2}-2 b^{2}\right )}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b \right )^{2}}+\frac {2 \left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (4 a^{4}-8 a^{2} b^{2}+4 b^{4}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{3}}}{d}\) | \(314\) |
| default | \(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {\frac {4 a^{2} b \left (4 a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {4 a \left (10 a^{4}+a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {4 a^{2} b \left (16 a^{2}-7 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}+\frac {4 a \,b^{2} \left (5 a^{2}-2 b^{2}\right )}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b \right )^{2}}+\frac {2 \left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (4 a^{4}-8 a^{2} b^{2}+4 b^{4}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{3}}}{d}\) | \(314\) |
| risch | \(\frac {x}{a^{3}}-\frac {i b^{2} \left (7 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}-4 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-17 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+8 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{4}-3 a^{2} b^{2}\right )}{\left (2 b \,{\mathrm e}^{i \left (d x +c \right )}-i a \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \right )^{2} \left (-a^{2}+b^{2}\right )^{2} d \,a^{3}}+\frac {3 b a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}-\frac {3 b a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}\) | \(680\) |
1/d*(2/a^3*arctan(tan(1/2*d*x+1/2*c))-2/a^3*b*(4*(1/8*a^2*b*(4*a^2-b^2)/(a ^4-2*a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)^3+1/8*a*(10*a^4+a^2*b^2-2*b^4)/(a^4-2 *a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)^2+1/8*a^2*b*(16*a^2-7*b^2)/(a^4-2*a^2*b^2 +b^4)*tan(1/2*d*x+1/2*c)+1/8*a*b^2*(5*a^2-2*b^2)/(a^4-2*a^2*b^2+b^4))/(tan (1/2*d*x+1/2*c)^2*b+2*a*tan(1/2*d*x+1/2*c)+b)^2+2*(6*a^4-5*a^2*b^2+2*b^4)/ (4*a^4-8*a^2*b^2+4*b^4)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*d*x+1/2*c )+2*a)/(-a^2+b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (161) = 322\).
Time = 0.30 (sec) , antiderivative size = 933, normalized size of antiderivative = 5.49 \[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=\left [\frac {4 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} - 4 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} d x - {\left (6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left (6 \, a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b \sin \left (d x + c\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (d x + c\right ) - 2 \, {\left (4 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d x - 3 \, {\left (2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right )} d \sin \left (d x + c\right ) - {\left (a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8}\right )} d\right )}}, \frac {2 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} d x - {\left (6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left (6 \, a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )}\right ) + {\left (5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (d x + c\right ) - {\left (4 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d x - 3 \, {\left (2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right )} d \sin \left (d x + c\right ) - {\left (a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8}\right )} d\right )}}\right ] \]
[1/4*(4*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cos(d*x + c)^2 - 4*(a^ 8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d*x - (6*a^6*b + a^4*b^3 - 3*a^2*b^5 + 2* b^7 - (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 5* a^3*b^4 + 2*a*b^6)*sin(d*x + c))*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(d* x + c)^2 + 2*a*b*sin(d*x + c) + a^2 + b^2 + 2*(b*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c))*sqrt(a^2 - b^2))/(a^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 2*(5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(d*x + c) - 2*(4 *(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x - 3*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5* b^6)*d*cos(d*x + c)^2 - 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*sin (d*x + c) - (a^11 - 2*a^9*b^2 + 2*a^5*b^6 - a^3*b^8)*d), 1/2*(2*(a^8 - 3*a ^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cos(d*x + c)^2 - 2*(a^8 - 2*a^6*b^2 + 2* a^2*b^6 - b^8)*d*x - (6*a^6*b + a^4*b^3 - 3*a^2*b^5 + 2*b^7 - (6*a^6*b - 5 *a^4*b^3 + 2*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6) *sin(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(d*x + c) + a)/((a^2 - b^2)*cos(d*x + c))) + (5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(d* x + c) - (4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x - 3*(2*a^6*b^2 - 3 *a^4*b^4 + a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7 *b^4 - a^5*b^6)*d*cos(d*x + c)^2 - 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4 *b^7)*d*sin(d*x + c) - (a^11 - 2*a^9*b^2 + 2*a^5*b^6 - a^3*b^8)*d)]
\[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \csc {\left (c + d x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=-\frac {\frac {{\left (6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {4 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2} b^{3} - 2 \, b^{5}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b\right )}^{2}} - \frac {d x + c}{a^{3}}}{d} \]
-((6*a^4*b - 5*a^2*b^3 + 2*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*d*x + 1/2*c) + a)/sqrt(-a^2 + b^2)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(-a^2 + b^2)) + (4*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - a*b^4*t an(1/2*d*x + 1/2*c)^3 + 10*a^4*b*tan(1/2*d*x + 1/2*c)^2 + a^2*b^3*tan(1/2* d*x + 1/2*c)^2 - 2*b^5*tan(1/2*d*x + 1/2*c)^2 + 16*a^3*b^2*tan(1/2*d*x + 1 /2*c) - 7*a*b^4*tan(1/2*d*x + 1/2*c) + 5*a^2*b^3 - 2*b^5)/((a^6 - 2*a^4*b^ 2 + a^2*b^4)*(b*tan(1/2*d*x + 1/2*c)^2 + 2*a*tan(1/2*d*x + 1/2*c) + b)^2) - (d*x + c)/a^3)/d
Time = 27.17 (sec) , antiderivative size = 5917, normalized size of antiderivative = 34.81 \[ \int \frac {1}{(a+b \csc (c+d x))^3} \, dx=\text {Too large to display} \]
((2*b^5 - 5*a^2*b^3)/(a^2*(a^4 + b^4 - 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)*( 7*b^4 - 16*a^2*b^2))/(a*(a^4 + b^4 - 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^3*( b^4 - 4*a^2*b^2))/(a*(a^4 + b^4 - 2*a^2*b^2)) - (tan(c/2 + (d*x)/2)^2*(5*a ^2*b - 2*b^3)*(2*a^2 + b^2))/(a^2*(a^4 + b^4 - 2*a^2*b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(4*a^2 + 2*b^2) + b^2*tan(c/2 + (d*x)/2)^4 + b^2 + 4*a*b*tan(c/ 2 + (d*x)/2)^3 + 4*a*b*tan(c/2 + (d*x)/2))) + (2*atan((((8*(4*a^2*b^10 - 1 6*a^4*b^8 + 24*a^6*b^6 - 16*a^8*b^4 + 4*a^10*b^2))/(a^13 + a^5*b^8 - 4*a^7 *b^6 + 6*a^9*b^4 - 4*a^11*b^2) - (((8*(4*a^14*b + 2*a^6*b^9 - 4*a^8*b^7 + 6*a^10*b^5 - 8*a^12*b^3))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11 *b^2) - (((8*(4*a^8*b^10 - 16*a^10*b^8 + 24*a^12*b^6 - 16*a^14*b^4 + 4*a^1 6*b^2))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2) + (8*tan(c/2 + (d*x)/2)*(12*a^18*b - 8*a^8*b^11 + 44*a^10*b^9 - 96*a^12*b^7 + 104*a^14 *b^5 - 56*a^16*b^3))/(a^14 + a^6*b^8 - 4*a^8*b^6 + 6*a^10*b^4 - 4*a^12*b^2 ))*1i)/a^3 + (8*tan(c/2 + (d*x)/2)*(8*a^6*b^10 - 36*a^8*b^8 + 72*a^10*b^6 - 68*a^12*b^4 + 24*a^14*b^2))/(a^14 + a^6*b^8 - 4*a^8*b^6 + 6*a^10*b^4 - 4 *a^12*b^2))*1i)/a^3 + (8*tan(c/2 + (d*x)/2)*(8*a^12*b - 8*a^2*b^11 + 44*a^ 4*b^9 - 105*a^6*b^7 + 124*a^8*b^5 - 72*a^10*b^3))/(a^14 + a^6*b^8 - 4*a^8* b^6 + 6*a^10*b^4 - 4*a^12*b^2))/a^3 + ((8*(4*a^2*b^10 - 16*a^4*b^8 + 24*a^ 6*b^6 - 16*a^8*b^4 + 4*a^10*b^2))/(a^13 + a^5*b^8 - 4*a^7*b^6 + 6*a^9*b^4 - 4*a^11*b^2) + (((((8*(4*a^8*b^10 - 16*a^10*b^8 + 24*a^12*b^6 - 16*a^1...