Integrand size = 29, antiderivative size = 194 \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^3 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}+\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \] Output:
2*b^3*(a^2-b^2)^(1/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^5 /d+1/8*(a^4+4*a^2*b^2-8*b^4)*arctanh(cos(d*x+c))/a^5/d-1/3*b*(a^2-3*b^2)*c ot(d*x+c)/a^4/d+1/8*(a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)/a^3/d+1/3*b*cot(d*x+ c)*csc(d*x+c)^2/a^2/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(430\) vs. \(2(194)=388\).
Time = 7.49 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.22 \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^3 \sqrt {a^2-b^2} \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}+\frac {\left (-a^2 b \cos \left (\frac {1}{2} (c+d x)\right )+3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {\left (a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (-a^4-4 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (-a^2+4 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \] Input:
Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
(2*b^3*Sqrt[a^2 - b^2]*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Si n[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*d) + ((-(a^2*b*Cos[(c + d*x)/2]) + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + ((a^2 - 4*b^2)*Csc[ (c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a ^2*d) - Csc[(c + d*x)/2]^4/(64*a*d) + ((a^4 + 4*a^2*b^2 - 8*b^4)*Log[Cos[( c + d*x)/2]])/(8*a^5*d) + ((-a^4 - 4*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2] ])/(8*a^5*d) + ((-a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^3*d) + Sec[(c + d *x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[ (c + d*x)/2]))/(6*a^4*d) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2 *d)
Time = 1.79 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.16, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {3042, 3368, 3042, 3535, 25, 3042, 3534, 3042, 3534, 25, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(c+d x)\right ) \csc ^5(c+d x)}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-\sin (c+d x)^2}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {\csc ^4(c+d x) \left (-3 b \sin ^2(c+d x)+a \sin (c+d x)+4 b\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\csc ^4(c+d x) \left (-3 b \sin ^2(c+d x)+a \sin (c+d x)+4 b\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-3 b \sin (c+d x)^2+a \sin (c+d x)+4 b}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc ^3(c+d x) \left (8 b^2 \sin ^2(c+d x)-a b \sin (c+d x)+3 \left (a^2-4 b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {8 b^2 \sin (c+d x)^2-a b \sin (c+d x)+3 \left (a^2-4 b^2\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-3 b \left (a^2-4 b^2\right ) \sin ^2(c+d x)-a \left (3 a^2+4 b^2\right ) \sin (c+d x)+8 b \left (a^2-3 b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-3 b \left (a^2-4 b^2\right ) \sin ^2(c+d x)-a \left (3 a^2+4 b^2\right ) \sin (c+d x)+8 b \left (a^2-3 b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {\int \frac {-3 b \left (a^2-4 b^2\right ) \sin (c+d x)^2-a \left (3 a^2+4 b^2\right ) \sin (c+d x)+8 b \left (a^2-3 b^2\right )}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {-\frac {\frac {\int -\frac {3 \csc (c+d x) \left (a^4+4 b^2 a^2+b \left (a^2-4 b^2\right ) \sin (c+d x) a-8 b^4\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \int \frac {\csc (c+d x) \left (a^4+4 b^2 a^2+b \left (a^2-4 b^2\right ) \sin (c+d x) a-8 b^4\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \int \frac {a^4+4 b^2 a^2+b \left (a^2-4 b^2\right ) \sin (c+d x) a-8 b^4}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \left (\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^3 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \left (\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {8 b^3 \left (a^2-b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \left (\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^3 \left (a^2-b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \left (\frac {32 b^3 \left (a^2-b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {-\frac {-\frac {3 \left (\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^3 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {-\frac {3 \left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {-\frac {8 b \left (a^2-3 b^2\right ) \cot (c+d x)}{a d}-\frac {3 \left (-\frac {16 b^3 \sqrt {a^2-b^2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}}{2 a}}{3 a}-\frac {4 b \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{4 a}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\) |
Input:
Int[(Cot[c + d*x]^2*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
-1/4*(Cot[c + d*x]*Csc[c + d*x]^3)/(a*d) - ((-4*b*Cot[c + d*x]*Csc[c + d*x ]^2)/(3*a*d) + (-1/2*((-3*((-16*b^3*Sqrt[a^2 - b^2]*ArcTan[(2*b + 2*a*Tan[ (c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d) - ((a^4 + 4*a^2*b^2 - 8*b^4)*Arc Tanh[Cos[c + d*x]])/(a*d)))/a - (8*b*(a^2 - 3*b^2)*Cot[c + d*x])/(a*d))/a - (3*(a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a))/(4*a)
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[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[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*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 = 0.79 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}}{3}+2 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{4}}+\frac {2 b^{3} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\left (-2 a^{4}-8 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b^{2}}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(252\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{4}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}}{3}+2 a \,b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{16 a^{4}}+\frac {2 b^{3} \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\left (-2 a^{4}-8 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b^{2}}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(252\) |
| risch | \(\frac {i \left (21 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-24 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+24 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+24 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-72 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+21 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-8 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2} b +72 \,{\mathrm e}^{2 i \left (d x +c \right )} b^{3}+8 a^{2} b -24 b^{3}\right )}{12 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{5}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{a^{5} d}\) | \(486\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/16/a^4*(1/4*tan(1/2*d*x+1/2*c)^4*a^3-2/3*b*tan(1/2*d*x+1/2*c)^3*a^2 +2*a*b^2*tan(1/2*d*x+1/2*c)^2+2*a^2*b*tan(1/2*d*x+1/2*c)-8*tan(1/2*d*x+1/2 *c)*b^3)+2*b^3*(a^2-b^2)^(1/2)/a^5*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b) /(a^2-b^2)^(1/2))-1/64/a/tan(1/2*d*x+1/2*c)^4+1/16/a^5*(-2*a^4-8*a^2*b^2+1 6*b^4)*ln(tan(1/2*d*x+1/2*c))+1/24/a^2*b/tan(1/2*d*x+1/2*c)^3-1/8/a^3*b^2/ tan(1/2*d*x+1/2*c)^2-1/8*b*(a^2-4*b^2)/a^4/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (179) = 358\).
Time = 0.29 (sec) , antiderivative size = 808, normalized size of antiderivative = 4.16 \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
[-1/48*(6*(a^4 - 4*a^2*b^2)*cos(d*x + c)^3 - 24*(b^3*cos(d*x + c)^4 - 2*b^ 3*cos(d*x + c)^2 + b^3)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^ 2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*co s(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a ^2 - b^2)) + 6*(a^4 + 4*a^2*b^2)*cos(d*x + c) - 3*((a^4 + 4*a^2*b^2 - 8*b^ 4)*cos(d*x + c)^4 + a^4 + 4*a^2*b^2 - 8*b^4 - 2*(a^4 + 4*a^2*b^2 - 8*b^4)* cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 3*((a^4 + 4*a^2*b^2 - 8*b^4) *cos(d*x + c)^4 + a^4 + 4*a^2*b^2 - 8*b^4 - 2*(a^4 + 4*a^2*b^2 - 8*b^4)*co s(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 16*(3*a*b^3*cos(d*x + c) + (a ^3*b - 3*a*b^3)*cos(d*x + c)^3)*sin(d*x + c))/(a^5*d*cos(d*x + c)^4 - 2*a^ 5*d*cos(d*x + c)^2 + a^5*d), -1/48*(6*(a^4 - 4*a^2*b^2)*cos(d*x + c)^3 + 4 8*(b^3*cos(d*x + c)^4 - 2*b^3*cos(d*x + c)^2 + b^3)*sqrt(a^2 - b^2)*arctan (-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 6*(a^4 + 4*a^2*b^ 2)*cos(d*x + c) - 3*((a^4 + 4*a^2*b^2 - 8*b^4)*cos(d*x + c)^4 + a^4 + 4*a^ 2*b^2 - 8*b^4 - 2*(a^4 + 4*a^2*b^2 - 8*b^4)*cos(d*x + c)^2)*log(1/2*cos(d* x + c) + 1/2) + 3*((a^4 + 4*a^2*b^2 - 8*b^4)*cos(d*x + c)^4 + a^4 + 4*a^2* b^2 - 8*b^4 - 2*(a^4 + 4*a^2*b^2 - 8*b^4)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 16*(3*a*b^3*cos(d*x + c) + (a^3*b - 3*a*b^3)*cos(d*x + c)^3 )*sin(d*x + c))/(a^5*d*cos(d*x + c)^4 - 2*a^5*d*cos(d*x + c)^2 + a^5*d)]
\[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**3/(a+b*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)**2*csc(c + d*x)**3/(a + b*sin(c + d*x)), x)
Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c)),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 = 0.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.73 \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {24 \, {\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {384 \, {\left (a^{2} b^{3} - b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {50 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 200 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{4}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/192*((3*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 24 *a*b^2*tan(1/2*d*x + 1/2*c)^2 + 24*a^2*b*tan(1/2*d*x + 1/2*c) - 96*b^3*tan (1/2*d*x + 1/2*c))/a^4 - 24*(a^4 + 4*a^2*b^2 - 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 + 384*(a^2*b^3 - b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn (a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^ 2)*a^5) + (50*a^4*tan(1/2*d*x + 1/2*c)^4 + 200*a^2*b^2*tan(1/2*d*x + 1/2*c )^4 - 400*b^4*tan(1/2*d*x + 1/2*c)^4 - 24*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 9 6*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 24*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 8*a^3 *b*tan(1/2*d*x + 1/2*c) - 3*a^4)/(a^5*tan(1/2*d*x + 1/2*c)^4))/d
Time = 20.77 (sec) , antiderivative size = 873, normalized size of antiderivative = 4.50 \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)^2/(sin(c + d*x)^3*(a + b*sin(c + d*x))),x)
Output:
tan(c/2 + (d*x)/2)^4/(64*a*d) + (tan(c/2 + (d*x)/2)*(b/(8*a^2) - b^3/(2*a^ 4)))/d - (tan(c/2 + (d*x)/2)^3*(2*a^2*b - 8*b^3) + a^3/4 - (2*a^2*b*tan(c/ 2 + (d*x)/2))/3 + 2*a*b^2*tan(c/2 + (d*x)/2)^2)/(16*a^4*d*tan(c/2 + (d*x)/ 2)^4) - (log(tan(c/2 + (d*x)/2))*(a^4 - 8*b^4 + 4*a^2*b^2))/(8*a^5*d) - (b *tan(c/2 + (d*x)/2)^3)/(24*a^2*d) + (b^2*tan(c/2 + (d*x)/2)^2)/(8*a^3*d) + (b^3*atan(((b^3*(b^2 - a^2)^(1/2)*((tan(c/2 + (d*x)/2)*(a^9 + 32*a^3*b^6 - 32*a^5*b^4 + 2*a^7*b^2))/(4*a^7) - (a^9*b - 16*a^5*b^5 + 12*a^7*b^3)/(4* a^8) + (b^3*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^10 - 32*a^8*b^2))/(4*a^7) )*(b^2 - a^2)^(1/2))/a^5)*1i)/a^5 - (b^3*(b^2 - a^2)^(1/2)*((a^9*b - 16*a^ 5*b^5 + 12*a^7*b^3)/(4*a^8) - (tan(c/2 + (d*x)/2)*(a^9 + 32*a^3*b^6 - 32*a ^5*b^4 + 2*a^7*b^2))/(4*a^7) + (b^3*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^1 0 - 32*a^8*b^2))/(4*a^7))*(b^2 - a^2)^(1/2))/a^5)*1i)/a^5)/((8*b^9 - 12*a^ 2*b^7 + 3*a^4*b^5 + a^6*b^3)/(2*a^8) + (tan(c/2 + (d*x)/2)*(8*b^8 - 10*a^2 *b^6 + 2*a^4*b^4))/(2*a^7) + (b^3*(b^2 - a^2)^(1/2)*((tan(c/2 + (d*x)/2)*( a^9 + 32*a^3*b^6 - 32*a^5*b^4 + 2*a^7*b^2))/(4*a^7) - (a^9*b - 16*a^5*b^5 + 12*a^7*b^3)/(4*a^8) + (b^3*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^10 - 32* a^8*b^2))/(4*a^7))*(b^2 - a^2)^(1/2))/a^5))/a^5 + (b^3*(b^2 - a^2)^(1/2)*( (a^9*b - 16*a^5*b^5 + 12*a^7*b^3)/(4*a^8) - (tan(c/2 + (d*x)/2)*(a^9 + 32* a^3*b^6 - 32*a^5*b^4 + 2*a^7*b^2))/(4*a^7) + (b^3*(2*a^2*b - (tan(c/2 + (d *x)/2)*(24*a^10 - 32*a^8*b^2))/(4*a^7))*(b^2 - a^2)^(1/2))/a^5))/a^5))*...
Time = 0.19 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.28 \[ \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {48 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{4} b^{3}-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b +24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{4}-12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} b^{2}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b -6 \cos \left (d x +c \right ) a^{4}-3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{4}-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a^{2} b^{2}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} b^{4}}{24 \sin \left (d x +c \right )^{4} a^{5} d} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c)),x)
Output:
(48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin (c + d*x)**4*b**3 - 8*cos(c + d*x)*sin(c + d*x)**3*a**3*b + 24*cos(c + d*x )*sin(c + d*x)**3*a*b**3 + 3*cos(c + d*x)*sin(c + d*x)**2*a**4 - 12*cos(c + d*x)*sin(c + d*x)**2*a**2*b**2 + 8*cos(c + d*x)*sin(c + d*x)*a**3*b - 6* cos(c + d*x)*a**4 - 3*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4 - 12*log( tan((c + d*x)/2))*sin(c + d*x)**4*a**2*b**2 + 24*log(tan((c + d*x)/2))*sin (c + d*x)**4*b**4)/(24*sin(c + d*x)**4*a**5*d)