Integrand size = 27, antiderivative size = 157 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \left (2 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \sqrt {a^2-b^2} d}+\frac {\left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}+\frac {3 b \cot (c+d x)}{a^3 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))} \] Output:
2*b*(2*a^2-3*b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^4/(a^ 2-b^2)^(1/2)/d+1/2*(a^2-6*b^2)*arctanh(cos(d*x+c))/a^4/d+3*b*cot(d*x+c)/a^ 3/d-3/2*cot(d*x+c)*csc(d*x+c)/a^2/d+cot(d*x+c)*csc(d*x+c)/a/d/(a+b*sin(d*x +c))
Time = 3.90 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.25 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-\frac {16 b \left (-2 a^2+3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+8 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+4 \left (a^2-6 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \left (a^2-6 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {8 a b^2 \cos (c+d x)}{a+b \sin (c+d x)}-8 a b \tan \left (\frac {1}{2} (c+d x)\right )}{8 a^4 d} \] Input:
Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
((-16*b*(-2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]) /Sqrt[a^2 - b^2] + 8*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 4*(a^ 2 - 6*b^2)*Log[Cos[(c + d*x)/2]] - 4*(a^2 - 6*b^2)*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 + (8*a*b^2*Cos[c + d*x])/(a + b*Sin[c + d*x]) - 8* a*b*Tan[(c + d*x)/2])/(8*a^4*d)
Time = 1.54 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.43, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3368, 3042, 3535, 3042, 3535, 25, 3042, 3534, 3042, 3480, 3042, 3139, 1083, 217, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(c+d x)\right ) \csc ^3(c+d x)}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-\sin (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (3 \left (a^2-b^2\right )-2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 \left (a^2-b^2\right )-2 \left (a^2-b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-3 b \left (a^2-b^2\right ) \sin ^2(c+d x)+a \left (a^2-b^2\right ) \sin (c+d x)+6 b \left (a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-3 b \left (a^2-b^2\right ) \sin ^2(c+d x)+a \left (a^2-b^2\right ) \sin (c+d x)+6 b \left (a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-3 b \left (a^2-b^2\right ) \sin (c+d x)^2+a \left (a^2-b^2\right ) \sin (c+d x)+6 b \left (a^2-b^2\right )}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\csc (c+d x) \left (a^4-7 b^2 a^2-3 b \left (a^2-b^2\right ) \sin (c+d x) a+6 b^4\right )}{a+b \sin (c+d x)}dx}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {a^4-7 b^2 a^2-3 b \left (a^2-b^2\right ) \sin (c+d x) a+6 b^4}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (a^4-7 a^2 b^2+6 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {2 b \left (2 a^4-5 a^2 b^2+3 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (a^4-7 a^2 b^2+6 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {2 b \left (2 a^4-5 a^2 b^2+3 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (a^4-7 a^2 b^2+6 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {4 b \left (2 a^4-5 a^2 b^2+3 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\frac {\frac {8 b \left (2 a^4-5 a^2 b^2+3 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {\left (a^4-7 a^2 b^2+6 b^4\right ) \int \csc (c+d x)dx}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (a^4-7 a^2 b^2+6 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {4 b \left (2 a^4-5 a^2 b^2+3 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {3 \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\frac {-\frac {4 b \left (2 a^4-5 a^2 b^2+3 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {\left (a^4-7 a^2 b^2+6 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x)}{a d}}{2 a}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}\) |
Input:
Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
(-1/2*(((-4*b*(2*a^4 - 5*a^2*b^2 + 3*b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/ 2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - ((a^4 - 7*a^2*b^2 + 6*b^ 4)*ArcTanh[Cos[c + d*x]])/(a*d))/a - (6*b*(a^2 - b^2)*Cot[c + d*x])/(a*d)) /a - (3*(a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(a*(a^2 - b^2)) + (Cot[c + d*x]*Csc[c + d*x])/(a*d*(a + b*Sin[c + d*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 = 0.90 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 b \left (\frac {\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a b}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4}}}{d}\) | \(206\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 b \left (\frac {\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a b}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4}}}{d}\) | \(206\) |
| risch | \(\frac {2 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a b \,{\mathrm e}^{5 i \left (d x +c \right )}+12 i a b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 i a b \,{\mathrm e}^{i \left (d x +c \right )}+6 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) d \,a^{3}}-\frac {2 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{2}}+\frac {3 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{4}}+\frac {2 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{2}}-\frac {3 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{4} d}\) | \(532\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/4/a^3*(1/2*tan(1/2*d*x+1/2*c)^2*a-4*b*tan(1/2*d*x+1/2*c))-1/8/a^2/t an(1/2*d*x+1/2*c)^2+1/4*(-2*a^2+12*b^2)/a^4*ln(tan(1/2*d*x+1/2*c))+b/a^3/t an(1/2*d*x+1/2*c)+4*b/a^4*((1/2*b^2*tan(1/2*d*x+1/2*c)+1/2*a*b)/(tan(1/2*d *x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1/2*(2*a^2-3*b^2)/(a^2-b^2)^(1/2)* arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (148) = 296\).
Time = 0.28 (sec) , antiderivative size = 1130, normalized size of antiderivative = 7.20 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
[1/4*(12*(a^3*b^2 - a*b^4)*cos(d*x + c)^3 - 6*(a^4*b - a^2*b^3)*cos(d*x + c)*sin(d*x + c) - 2*(2*a^3*b - 3*a*b^3 - (2*a^3*b - 3*a*b^3)*cos(d*x + c)^ 2 + (2*a^2*b^2 - 3*b^4 - (2*a^2*b^2 - 3*b^4)*cos(d*x + c)^2)*sin(d*x + c)) *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*cos(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)) + 2*(a^5 - 7 *a^3*b^2 + 6*a*b^4)*cos(d*x + c) - (a^5 - 7*a^3*b^2 + 6*a*b^4 - (a^5 - 7*a ^3*b^2 + 6*a*b^4)*cos(d*x + c)^2 + (a^4*b - 7*a^2*b^3 + 6*b^5 - (a^4*b - 7 *a^2*b^3 + 6*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2 ) + (a^5 - 7*a^3*b^2 + 6*a*b^4 - (a^5 - 7*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^ 2 + (a^4*b - 7*a^2*b^3 + 6*b^5 - (a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^ 2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^7 - a^5*b^2)*d*cos(d*x + c)^2 - (a^7 - a^5*b^2)*d + ((a^6*b - a^4*b^3)*d*cos(d*x + c)^2 - (a^6*b - a^4*b^3)*d)*sin(d*x + c)), 1/4*(12*(a^3*b^2 - a*b^4)*cos(d*x + c)^3 - 6* (a^4*b - a^2*b^3)*cos(d*x + c)*sin(d*x + c) + 4*(2*a^3*b - 3*a*b^3 - (2*a^ 3*b - 3*a*b^3)*cos(d*x + c)^2 + (2*a^2*b^2 - 3*b^4 - (2*a^2*b^2 - 3*b^4)*c os(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/ (sqrt(a^2 - b^2)*cos(d*x + c))) + 2*(a^5 - 7*a^3*b^2 + 6*a*b^4)*cos(d*x + c) - (a^5 - 7*a^3*b^2 + 6*a*b^4 - (a^5 - 7*a^3*b^2 + 6*a*b^4)*cos(d*x + c) ^2 + (a^4*b - 7*a^2*b^3 + 6*b^5 - (a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x +...
\[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)/(a+b*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**2*csc(c + d*x)/(a + b*sin(c + d*x))**2, x)
Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)/(a+b*sin(d*x+c))^2,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 = 0.16 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {16 \, {\left (2 \, a^{2} b - 3 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {16 \, {\left (b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{4}} - \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
-1/8*(4*(a^2 - 6*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - 16*(2*a^2*b - 3 *b^3)*(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^4) - (a^2*tan(1/2*d*x + 1/ 2*c)^2 - 8*a*b*tan(1/2*d*x + 1/2*c))/a^4 - 16*(b^3*tan(1/2*d*x + 1/2*c) + a*b^2)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^4) - ( 6*a^2*tan(1/2*d*x + 1/2*c)^2 - 36*b^2*tan(1/2*d*x + 1/2*c)^2 + 8*a*b*tan(1 /2*d*x + 1/2*c) - a^2)/(a^4*tan(1/2*d*x + 1/2*c)^2))/d
Time = 33.91 (sec) , antiderivative size = 966, normalized size of antiderivative = 6.15 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)^2/(sin(c + d*x)*(a + b*sin(c + d*x))^2),x)
Output:
tan(c/2 + (d*x)/2)^2/(8*a^2*d) - (tan(c/2 + (d*x)/2)^2*(a^2/2 - 16*b^2) + a^2/2 - (4*tan(c/2 + (d*x)/2)^3*(a^2*b + 2*b^3))/a - 3*a*b*tan(c/2 + (d*x) /2))/(d*(4*a^4*tan(c/2 + (d*x)/2)^2 + 4*a^4*tan(c/2 + (d*x)/2)^4 + 8*a^3*b *tan(c/2 + (d*x)/2)^3)) - (b*tan(c/2 + (d*x)/2))/(a^3*d) - (log(tan(c/2 + (d*x)/2))*(a^2 - 6*b^2))/(2*a^4*d) - (b*atan(((b*(-(a + b)*(a - b))^(1/2)* (2*a^2 - 3*b^2)*((5*a^6*b - 12*a^4*b^3)/a^6 - (tan(c/2 + (d*x)/2)*(a^6 + 2 4*a^2*b^4 - 16*a^4*b^2))/a^5 + (b*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan (c/2 + (d*x)/2)*(6*a^8 - 8*a^6*b^2))/a^5)*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2) )*1i)/(a^6 - a^4*b^2) - (b*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((tan( c/2 + (d*x)/2)*(a^6 + 24*a^2*b^4 - 16*a^4*b^2))/a^5 - (5*a^6*b - 12*a^4*b^ 3)/a^6 + (b*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^8 - 8*a^6*b^2))/a^5)*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2))*1i)/(a^6 - a^4*b^2)) /((2*(2*a^4*b + 18*b^5 - 15*a^2*b^3))/a^6 + (2*tan(c/2 + (d*x)/2)*(18*b^4 - 12*a^2*b^2))/a^5 + (b*(-(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((5*a^6*b - 12*a^4*b^3)/a^6 - (tan(c/2 + (d*x)/2)*(a^6 + 24*a^2*b^4 - 16*a^4*b^2))/ a^5 + (b*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^8 - 8*a^6*b^2))/a^5)*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2)))/(a^6 - a^4*b^2) + (b*( -(a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((tan(c/2 + (d*x)/2)*(a^6 + 24*a^2 *b^4 - 16*a^4*b^2))/a^5 - (5*a^6*b - 12*a^4*b^3)/a^6 + (b*(-(a + b)*(a - b ))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(6*a^8 - 8*a^6*b^2))/a^5)*(2*a^...
Time = 0.19 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.22 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {8 \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 )^{3} a^{2} b^{2}-12 \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 )^{3} b^{4}+8 \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 )^{2} a^{3} b -12 \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 )^{2} a \,b^{3}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{2}-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{4}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b -3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}-\cos \left (d x +c \right ) a^{5}+\cos \left (d x +c \right ) a^{3} b^{2}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3} a^{4} b +7 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3} a^{2} b^{3}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3} b^{5}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a^{5}+7 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a^{3} b^{2}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a \,b^{4}}{2 \sin \left (d x +c \right )^{2} a^{4} d \left (\sin \left (d x +c \right ) a^{2} b -\sin \left (d x +c \right ) b^{3}+a^{3}-a \,b^{2}\right )} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)/(a+b*sin(d*x+c))^2,x)
Output:
(8*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin( c + d*x)**3*a**2*b**2 - 12*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b) /sqrt(a**2 - b**2))*sin(c + d*x)**3*b**4 + 8*sqrt(a**2 - b**2)*atan((tan(( c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**3*b - 12*sqrt(a** 2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2 *a*b**3 + 6*cos(c + d*x)*sin(c + d*x)**2*a**3*b**2 - 6*cos(c + d*x)*sin(c + d*x)**2*a*b**4 + 3*cos(c + d*x)*sin(c + d*x)*a**4*b - 3*cos(c + d*x)*sin (c + d*x)*a**2*b**3 - cos(c + d*x)*a**5 + cos(c + d*x)*a**3*b**2 - log(tan ((c + d*x)/2))*sin(c + d*x)**3*a**4*b + 7*log(tan((c + d*x)/2))*sin(c + d* x)**3*a**2*b**3 - 6*log(tan((c + d*x)/2))*sin(c + d*x)**3*b**5 - log(tan(( c + d*x)/2))*sin(c + d*x)**2*a**5 + 7*log(tan((c + d*x)/2))*sin(c + d*x)** 2*a**3*b**2 - 6*log(tan((c + d*x)/2))*sin(c + d*x)**2*a*b**4)/(2*sin(c + d *x)**2*a**4*d*(sin(c + d*x)*a**2*b - sin(c + d*x)*b**3 + a**3 - a*b**2))