Integrand size = 13, antiderivative size = 241 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{5/2}}-\frac {\left (a^2+12 b^2\right ) \text {arctanh}(\cos (x))}{2 a^5}+\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{2 a^4 \left (a^2-b^2\right )^2}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{2 a^3 \left (a^2-b^2\right )^2}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
-b^3*(20*a^4-29*a^2*b^2+12*b^4)*arctan((b+a*tan(1/2*x))/(a^2-b^2)^(1/2))/a ^5/(a^2-b^2)^(5/2)-1/2*(a^2+12*b^2)*arctanh(cos(x))/a^5+3/2*b*(2*a^4-7*a^2 *b^2+4*b^4)*cot(x)/a^4/(a^2-b^2)^2-1/2*(a^4-10*a^2*b^2+6*b^4)*cot(x)*csc(x )/a^3/(a^2-b^2)^2-1/2*b^2*cot(x)*csc(x)/a/(a^2-b^2)/(a+b*sin(x))^2-1/2*b^2 *(7*a^2-4*b^2)*cot(x)*csc(x)/a^2/(a^2-b^2)^2/(a+b*sin(x))
Time = 1.51 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.91 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=\frac {-\frac {8 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+12 a b \cot \left (\frac {x}{2}\right )-a^2 \csc ^2\left (\frac {x}{2}\right )-4 \left (a^2+12 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \left (a^2+12 b^2\right ) \log \left (\sin \left (\frac {x}{2}\right )\right )+a^2 \sec ^2\left (\frac {x}{2}\right )-\frac {4 a^2 b^4 \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac {12 a b^4 \left (-3 a^2+2 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-12 a b \tan \left (\frac {x}{2}\right )}{8 a^5} \]
((-8*b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 12*a*b*Cot[x/2] - a^2*Csc[x/2]^2 - 4*(a^2 + 12 *b^2)*Log[Cos[x/2]] + 4*(a^2 + 12*b^2)*Log[Sin[x/2]] + a^2*Sec[x/2]^2 - (4 *a^2*b^4*Cos[x])/((a - b)*(a + b)*(a + b*Sin[x])^2) + (12*a*b^4*(-3*a^2 + 2*b^2)*Cos[x])/((a - b)^2*(a + b)^2*(a + b*Sin[x])) - 12*a*b*Tan[x/2])/(8* a^5)
Time = 1.74 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.308, Rules used = {3042, 3281, 3042, 3534, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x)^3 (a+b \sin (x))^3}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int \frac {\csc ^3(x) \left (3 b^2 \sin ^2(x)-2 a b \sin (x)+2 \left (a^2-2 b^2\right )\right )}{(a+b \sin (x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 b^2 \sin (x)^2-2 a b \sin (x)+2 \left (a^2-2 b^2\right )}{\sin (x)^3 (a+b \sin (x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^3(x) \left (2 b^2 \left (7 a^2-4 b^2\right ) \sin ^2(x)-a b \left (4 a^2-b^2\right ) \sin (x)+2 \left (a^4-10 b^2 a^2+6 b^4\right )\right )}{a+b \sin (x)}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 b^2 \left (7 a^2-4 b^2\right ) \sin (x)^2-a b \left (4 a^2-b^2\right ) \sin (x)+2 \left (a^4-10 b^2 a^2+6 b^4\right )}{\sin (x)^3 (a+b \sin (x))}dx}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {2 \csc ^2(x) \left (-b \left (a^4-10 b^2 a^2+6 b^4\right ) \sin ^2(x)-a \left (a^4+4 b^2 a^2-2 b^4\right ) \sin (x)+3 b \left (2 a^4-7 b^2 a^2+4 b^4\right )\right )}{a+b \sin (x)}dx}{2 a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\csc ^2(x) \left (-b \left (a^4-10 b^2 a^2+6 b^4\right ) \sin ^2(x)-a \left (a^4+4 b^2 a^2-2 b^4\right ) \sin (x)+3 b \left (2 a^4-7 b^2 a^2+4 b^4\right )\right )}{a+b \sin (x)}dx}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-b \left (a^4-10 b^2 a^2+6 b^4\right ) \sin (x)^2-a \left (a^4+4 b^2 a^2-2 b^4\right ) \sin (x)+3 b \left (2 a^4-7 b^2 a^2+4 b^4\right )}{\sin (x)^2 (a+b \sin (x))}dx}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int -\frac {\csc (x) \left (\left (a^2+12 b^2\right ) \left (a^2-b^2\right )^2+a b \left (a^4-10 b^2 a^2+6 b^4\right ) \sin (x)\right )}{a+b \sin (x)}dx}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {\csc (x) \left (\left (a^2+12 b^2\right ) \left (a^2-b^2\right )^2+a b \left (a^4-10 b^2 a^2+6 b^4\right ) \sin (x)\right )}{a+b \sin (x)}dx}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {\left (a^2+12 b^2\right ) \left (a^2-b^2\right )^2+a b \left (a^4-10 b^2 a^2+6 b^4\right ) \sin (x)}{\sin (x) (a+b \sin (x))}dx}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2+12 b^2\right ) \int \csc (x)dx}{a}-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \int \frac {1}{a+b \sin (x)}dx}{a}}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2+12 b^2\right ) \int \csc (x)dx}{a}-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \int \frac {1}{a+b \sin (x)}dx}{a}}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2+12 b^2\right ) \int \csc (x)dx}{a}-\frac {2 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {x}{2}\right )+2 b \tan \left (\frac {x}{2}\right )+a}d\tan \left (\frac {x}{2}\right )}{a}}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2+12 b^2\right ) \int \csc (x)dx}{a}+\frac {4 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{a}}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2+12 b^2\right ) \int \csc (x)dx}{a}-\frac {2 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {-\frac {\left (a^2-b^2\right )^2 \left (a^2+12 b^2\right ) \text {arctanh}(\cos (x))}{a}-\frac {2 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}}{a}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \cot (x)}{a}}{a}-\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cot (x) \csc (x)}{a}}{a \left (a^2-b^2\right )}-\frac {b^2 \left (7 a^2-4 b^2\right ) \cot (x) \csc (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}}{2 a \left (a^2-b^2\right )}-\frac {b^2 \cot (x) \csc (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}\) |
-1/2*(b^2*Cot[x]*Csc[x])/(a*(a^2 - b^2)*(a + b*Sin[x])^2) + ((-((-(((-2*b^ 3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(2*b + 2*a*Tan[x/2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]) - ((a^2 - b^2)^2*(a^2 + 12*b^2)*ArcTanh[Cos[x ]])/a)/a) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Cot[x])/a)/a) - ((a^4 - 10*a^ 2*b^2 + 6*b^4)*Cot[x]*Csc[x])/a)/(a*(a^2 - b^2)) - (b^2*(7*a^2 - 4*b^2)*Co t[x]*Csc[x])/(a*(a^2 - b^2)*(a + b*Sin[x])))/(2*a*(a^2 - b^2))
3.3.1.3.1 Defintions of rubi rules used
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_.)*(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 = 1.14 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(\frac {\frac {a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-6 b \tan \left (\frac {x}{2}\right )}{4 a^{4}}-\frac {1}{8 a^{3} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (2 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {x}{2}\right )}-\frac {4 b^{3} \left (\frac {\frac {a \,b^{2} \left (11 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}+\frac {b \left (10 a^{4}+13 a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}+\frac {a \,b^{2} \left (29 a^{2}-20 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}+\frac {a^{2} b \left (10 a^{2}-7 b^{2}\right )}{4 a^{4}-8 a^{2} b^{2}+4 b^{4}}}{{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-29 a^{2} b^{2}+12 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{a^{5}}\) | \(329\) |
| risch | \(\frac {-a^{5} b^{2} {\mathrm e}^{7 i x}+10 a^{3} b^{4} {\mathrm e}^{7 i x}-6 a \,b^{6} {\mathrm e}^{7 i x}+17 a^{5} b^{2} {\mathrm e}^{5 i x}+6 i a^{4} b^{3}-21 i a^{2} b^{5}-55 a^{5} b^{2} {\mathrm e}^{3 i x}+162 a^{3} b^{4} {\mathrm e}^{3 i x}-90 a \,b^{6} {\mathrm e}^{3 i x}-90 a^{3} b^{4} {\mathrm e}^{5 i x}+54 a \,b^{6} {\mathrm e}^{5 i x}+23 a^{5} b^{2} {\mathrm e}^{i x}-74 a^{3} b^{4} {\mathrm e}^{i x}+42 a \,b^{6} {\mathrm e}^{i x}-12 i b^{7} {\mathrm e}^{6 i x}+36 i b^{7} {\mathrm e}^{4 i x}-36 i b^{7} {\mathrm e}^{2 i x}+20 i a^{4} b^{3} {\mathrm e}^{6 i x}+5 i a^{2} b^{5} {\mathrm e}^{6 i x}-66 i a^{4} b^{3} {\mathrm e}^{4 i x}-15 i a^{2} b^{5} {\mathrm e}^{4 i x}+31 i b^{5} a^{2} {\mathrm e}^{2 i x}-4 i a^{6} b \,{\mathrm e}^{6 i x}+24 i a^{6} b \,{\mathrm e}^{4 i x}-20 i a^{6} b \,{\mathrm e}^{2 i x}+40 i a^{4} b^{3} {\mathrm e}^{2 i x}+12 i b^{7}+4 a^{7} {\mathrm e}^{5 i x}+4 a^{7} {\mathrm e}^{3 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} \left (a^{2}-b^{2}\right )^{2} a^{4}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}-\frac {6 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{a^{5}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{a^{5}}-\frac {10 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 )^{2} \left (a -b \right )^{2} a}+\frac {29 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 )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {6 i b^{7} \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^{5}}+\frac {10 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 )^{2} \left (a -b \right )^{2} a}-\frac {29 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 )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}+\frac {6 i b^{7} \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^{5}}\) | \(922\) |
1/4/a^4*(1/2*a*tan(1/2*x)^2-6*b*tan(1/2*x))-1/8/a^3/tan(1/2*x)^2+1/4/a^5*( 2*a^2+24*b^2)*ln(tan(1/2*x))+3/2/a^4*b/tan(1/2*x)-4*b^3/a^5*((1/4*a*b^2*(1 1*a^2-8*b^2)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^3+1/4*b*(10*a^4+13*a^2*b^2-14* b^4)/(a^4-2*a^2*b^2+b^4)*tan(1/2*x)^2+1/4*a*b^2*(29*a^2-20*b^2)/(a^4-2*a^2 *b^2+b^4)*tan(1/2*x)+1/4*a^2*b*(10*a^2-7*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+1/4*(20*a^4-29*a^2*b^2+12*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)))
Leaf count of result is larger than twice the leaf count of optimal. 971 vs. \(2 (225) = 450\).
Time = 1.53 (sec) , antiderivative size = 2005, normalized size of antiderivative = 8.32 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]
[-1/4*(2*(11*a^8*b^2 - 43*a^6*b^4 + 50*a^4*b^6 - 18*a^2*b^8)*cos(x)^3 + (2 0*a^6*b^3 - 9*a^4*b^5 - 17*a^2*b^7 + 12*b^9 + (20*a^4*b^5 - 29*a^2*b^7 + 1 2*b^9)*cos(x)^4 - (20*a^6*b^3 + 11*a^4*b^5 - 46*a^2*b^7 + 24*b^9)*cos(x)^2 + 2*(20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8 - (20*a^5*b^4 - 29*a^3*b^6 + 12*a *b^8)*cos(x)^2)*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^10 - 14*a^8*b^2 + 46*a^6 *b^4 - 51*a^4*b^6 + 18*a^2*b^8)*cos(x) + (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35 *a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^ 6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 3 5*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a *b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^10 + 10*a^8*b^2 - 24*a^ 6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b ^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31 *a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5* b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2) + 2*(3*(2*a^7*b^3 - 9*a^5*b^5 + 11*a^3*b^7 - 4*a*b^9)*cos(x)^3 - (4*a^9*b - 6*a^7*b^3 - 15*a^5 *b^5 + 29*a^3*b^7 - 12*a*b^9)*cos(x))*sin(x))/(a^13 - 2*a^11*b^2 + 2*a^...
\[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (225) = 450\).
Time = 0.33 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.13 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=-\frac {{\left (20 \, a^{4} b^{3} - 29 \, a^{2} b^{5} + 12 \, b^{7}\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^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, a^{8} \tan \left (\frac {1}{2} \, x\right )^{6} + 20 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, x\right )^{6} - 46 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, x\right )^{6} + 24 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, x\right )^{6} - 4 \, a^{7} b \tan \left (\frac {1}{2} \, x\right )^{5} + 104 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, x\right )^{5} - 108 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, x\right )^{5} + 32 \, a b^{7} \tan \left (\frac {1}{2} \, x\right )^{5} + 5 \, a^{8} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 165 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} - 80 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, x\right )^{4} - 16 \, b^{8} \tan \left (\frac {1}{2} \, x\right )^{4} - 12 \, a^{7} b \tan \left (\frac {1}{2} \, x\right )^{3} + 72 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 124 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} - 112 \, a b^{7} \tan \left (\frac {1}{2} \, x\right )^{3} + 4 \, a^{8} \tan \left (\frac {1}{2} \, x\right )^{2} - 28 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 124 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 76 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a^{7} b \tan \left (\frac {1}{2} \, x\right ) + 16 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 8 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, x\right ) + a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}}{8 \, {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\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 )}^{2}} + \frac {{\left (a^{2} + 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{5}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{6}} \]
-(20*a^4*b^3 - 29*a^2*b^5 + 12*b^7)*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arc tan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))/((a^9 - 2*a^7*b^2 + a^5*b^4)*sqrt (a^2 - b^2)) - 1/8*(2*a^8*tan(1/2*x)^6 + 20*a^6*b^2*tan(1/2*x)^6 - 46*a^4* b^4*tan(1/2*x)^6 + 24*a^2*b^6*tan(1/2*x)^6 - 4*a^7*b*tan(1/2*x)^5 + 104*a^ 5*b^3*tan(1/2*x)^5 - 108*a^3*b^5*tan(1/2*x)^5 + 32*a*b^7*tan(1/2*x)^5 + 5* a^8*tan(1/2*x)^4 - 2*a^6*b^2*tan(1/2*x)^4 + 165*a^4*b^4*tan(1/2*x)^4 - 80* a^2*b^6*tan(1/2*x)^4 - 16*b^8*tan(1/2*x)^4 - 12*a^7*b*tan(1/2*x)^3 + 72*a^ 5*b^3*tan(1/2*x)^3 + 124*a^3*b^5*tan(1/2*x)^3 - 112*a*b^7*tan(1/2*x)^3 + 4 *a^8*tan(1/2*x)^2 - 28*a^6*b^2*tan(1/2*x)^2 + 124*a^4*b^4*tan(1/2*x)^2 - 7 6*a^2*b^6*tan(1/2*x)^2 - 8*a^7*b*tan(1/2*x) + 16*a^5*b^3*tan(1/2*x) - 8*a^ 3*b^5*tan(1/2*x) + a^8 - 2*a^6*b^2 + a^4*b^4)/((a^9 - 2*a^7*b^2 + a^5*b^4) *(a*tan(1/2*x)^3 + 2*b*tan(1/2*x)^2 + a*tan(1/2*x))^2) + 1/2*(a^2 + 12*b^2 )*log(abs(tan(1/2*x)))/a^5 + 1/8*(a^3*tan(1/2*x)^2 - 12*a^2*b*tan(1/2*x))/ a^6
Time = 9.30 (sec) , antiderivative size = 2405, normalized size of antiderivative = 9.98 \[ \int \frac {\csc ^3(x)}{(a+b \sin (x))^3} \, dx=\text {Too large to display} \]
(4*a^2*b*tan(x/2) - a^3/2 + (tan(x/2)^2*(50*a*b^6 - a^7 - 85*a^3*b^4 + 24* a^5*b^2))/(a^4 + b^4 - 2*a^2*b^2) + (2*tan(x/2)^5*(3*a^6*b + 16*b^7 - 19*a ^2*b^5 - 6*a^4*b^3))/(a^4 + b^4 - 2*a^2*b^2) + (2*tan(x/2)^3*(5*a^6*b + 52 *b^7 - 77*a^2*b^5 + 2*a^4*b^3))/(a^4 + b^4 - 2*a^2*b^2) - (tan(x/2)^4*(a^8 - 112*b^8 + 56*a^2*b^6 + 177*a^4*b^4 - 50*a^6*b^2))/(2*a*(a^4 + b^4 - 2*a ^2*b^2)))/(tan(x/2)^4*(8*a^6 + 16*a^4*b^2) + 4*a^6*tan(x/2)^2 + 4*a^6*tan( x/2)^6 + 16*a^5*b*tan(x/2)^3 + 16*a^5*b*tan(x/2)^5) + tan(x/2)^2/(8*a^3) + (log(tan(x/2))*(a^2 + 12*b^2))/(2*a^5) - (3*b*tan(x/2))/(2*a^4) + (b^3*at an(((b^3*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2)*((a^1 1*b + 24*a^5*b^7 - 52*a^7*b^5 + 30*a^9*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(a^15 - 48*a^3*b^12 + 212*a^5*b^10 - 363*a^7*b^8 + 290*a^9*b^6 - 98*a^11*b^4 + 6*a^13*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4 *a^13*b^2) + (b^3*((2*a^14*b + 2*a^10*b^5 - 4*a^12*b^3)/(a^12 + a^8*b^4 - 2*a^10*b^2) - (tan(x/2)*(6*a^18 - 8*a^8*b^10 + 38*a^10*b^8 - 72*a^12*b^6 + 68*a^14*b^4 - 32*a^16*b^2))/(a^15 + a^7*b^8 - 4*a^9*b^6 + 6*a^11*b^4 - 4* a^13*b^2))*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2))/(2 *(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2)))*1 i)/(2*(a^15 - a^5*b^10 + 5*a^7*b^8 - 10*a^9*b^6 + 10*a^11*b^4 - 5*a^13*b^2 )) - (b^3*(-(a + b)^5*(a - b)^5)^(1/2)*(20*a^4 + 12*b^4 - 29*a^2*b^2)*((ta n(x/2)*(a^15 - 48*a^3*b^12 + 212*a^5*b^10 - 363*a^7*b^8 + 290*a^9*b^6 -...