Integrand size = 21, antiderivative size = 202 \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\left (2 a^4-9 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{3/2} d}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \] Output:
-(2*a^4-9*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/ a^4/(a^2-b^2)^(3/2)/d+3*b*arctanh(cos(d*x+c))/a^4/d-1/2*(5*a^2-6*b^2)*cot( d*x+c)/a^3/(a^2-b^2)/d+1/2*cot(d*x+c)/a/d/(a+b*sin(d*x+c))^2+1/2*(2*a^2-3* b^2)*cot(d*x+c)/a^2/(a^2-b^2)/d/(a+b*sin(d*x+c))
Time = 4.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {2 \left (2 a^4-9 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-a \cot \left (\frac {1}{2} (c+d x)\right )+6 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {a^2 b \cos (c+d x)}{(a+b \sin (c+d x))^2}+\frac {a b \left (-3 a^2+4 b^2\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))}+a \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \] Input:
Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]
Output:
((-2*(2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - a*Cot[(c + d*x)/2] + 6*b*Log[Cos[(c + d*x)/2] ] - 6*b*Log[Sin[(c + d*x)/2]] - (a^2*b*Cos[c + d*x])/(a + b*Sin[c + d*x])^ 2 + (a*b*(-3*a^2 + 4*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d* x])) + a*Tan[(c + d*x)/2])/(2*a^4*d)
Time = 1.61 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.23, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3202, 3042, 3535, 3042, 3535, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^2 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3202 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(c+d x)\right ) \csc ^2(c+d x)}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-\sin (c+d x)^2}{\sin (c+d x)^2 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int \frac {\csc ^2(c+d x) \left (3 \left (a^2-b^2\right )-2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 \left (a^2-b^2\right )-2 \left (a^2-b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^2 (a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^2(c+d x) \left (5 a^4-11 b^2 a^2-b \left (a^2-b^2\right ) \sin (c+d x) a+6 b^4-\left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {5 a^4-11 b^2 a^2-b \left (a^2-b^2\right ) \sin (c+d x) a+6 b^4-\left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {\csc (c+d x) \left (6 b \left (a^2-b^2\right )^2+a \left (2 a^4-5 b^2 a^2+3 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\csc (c+d x) \left (6 b \left (a^2-b^2\right )^2+a \left (2 a^4-5 b^2 a^2+3 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {6 b \left (a^2-b^2\right )^2+a \left (2 a^4-5 b^2 a^2+3 b^4\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {-\frac {\frac {6 b \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}+\frac {\left (2 a^6-11 a^4 b^2+15 a^2 b^4-6 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {6 b \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}+\frac {\left (2 a^6-11 a^4 b^2+15 a^2 b^4-6 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {-\frac {\frac {6 b \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}+\frac {2 \left (2 a^6-11 a^4 b^2+15 a^2 b^4-6 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {\frac {6 b \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}-\frac {4 \left (2 a^6-11 a^4 b^2+15 a^2 b^4-6 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {\frac {6 b \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}+\frac {2 \left (2 a^6-11 a^4 b^2+15 a^2 b^4-6 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac {-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x)}{a d}-\frac {\frac {2 \left (2 a^6-11 a^4 b^2+15 a^2 b^4-6 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {6 b \left (a^2-b^2\right )^2 \text {arctanh}(\cos (c+d x))}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
Input:
Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]
Output:
Cot[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((-(((2*(2*a^6 - 11*a^4*b^2 + 15*a^2*b^4 - 6*b^6)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^ 2])])/(a*Sqrt[a^2 - b^2]*d) - (6*b*(a^2 - b^2)^2*ArcTanh[Cos[c + d*x]])/(a *d))/a) - ((5*a^4 - 11*a^2*b^2 + 6*b^4)*Cot[c + d*x])/(a*d))/(a*(a^2 - b^2 )) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(a*d*(a + b*Sin[c + d*x])))/(2*a*(a^2 - b^2))
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_)/tan[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*((1 - Sin[e + f*x]^2)/Sin[e + f*x]^ 2), x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 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[((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 = 3.79 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {2 \left (\frac {\frac {a \,b^{2} \left (5 a^{2}-6 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a^{2}-2 b^{2}}+\frac {b \left (4 a^{4}+3 b^{2} a^{2}-10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}-2 b^{2}}+\frac {b^{2} a \left (11 a^{2}-14 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (4 a^{2}-5 b^{2}\right )}{2 a^{2}-2 b^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{4}-9 b^{2} a^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(299\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {2 \left (\frac {\frac {a \,b^{2} \left (5 a^{2}-6 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a^{2}-2 b^{2}}+\frac {b \left (4 a^{4}+3 b^{2} a^{2}-10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}-2 b^{2}}+\frac {b^{2} a \left (11 a^{2}-14 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (4 a^{2}-5 b^{2}\right )}{2 a^{2}-2 b^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{4}-9 b^{2} a^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{4}}-\frac {1}{2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(299\) |
| risch | \(\frac {i \left (-2 i a^{3} b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i a \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+20 i b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-24 i b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-3 b^{2} {\mathrm e}^{4 i \left (d x +c \right )} a^{2}-6 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+21 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+5 b^{2} a^{2}-6 b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (-i {\mathrm e}^{2 i \left (d x +c \right )} b +i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) a^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{4} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}\) | \(800\) |
Input:
int(cot(d*x+c)^2/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2/a^3*tan(1/2*d*x+1/2*c)-2/a^4*((1/2*a*b^2*(5*a^2-6*b^2)/(a^2-b^2)* tan(1/2*d*x+1/2*c)^3+1/2*b*(4*a^4+3*a^2*b^2-10*b^4)/(a^2-b^2)*tan(1/2*d*x+ 1/2*c)^2+1/2*b^2*a*(11*a^2-14*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)+1/2*a^2*b* (4*a^2-5*b^2)/(a^2-b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a) ^2+1/2*(2*a^4-9*a^2*b^2+6*b^4)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x +1/2*c)+2*b)/(a^2-b^2)^(1/2)))-1/2/a^3/tan(1/2*d*x+1/2*c)-3/a^4*b*ln(tan(1 /2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (191) = 382\).
Time = 0.39 (sec) , antiderivative size = 1394, normalized size of antiderivative = 6.90 \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
[-1/4*(2*(5*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^3 - 2*(8*a^6*b - 17*a^4*b^3 + 9*a^2*b^5)*cos(d*x + c)*sin(d*x + c) + (4*a^5*b - 18*a^3*b^3 + 12*a*b^5 - 2*(2*a^5*b - 9*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 7 *a^4*b^2 - 3*a^2*b^4 + 6*b^6 - (2*a^4*b^2 - 9*a^2*b^4 + 6*b^6)*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*(2*a^7 + a^5*b^2 - 9*a^3*b^4 + 6*a*b^6)*cos(d*x + c) + 6*(2*a^5*b ^2 - 4*a^3*b^4 + 2*a*b^6 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + (a^6*b - a^4*b^3 - a^2*b^5 + b^7 - (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 6*(2*a^5*b^2 - 4*a^3*b^ 4 + 2*a*b^6 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + (a^6*b - a^ 4*b^3 - a^2*b^5 + b^7 - (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d* x + c))*log(-1/2*cos(d*x + c) + 1/2))/(2*(a^9*b - 2*a^7*b^3 + a^5*b^5)*d*c os(d*x + c)^2 - 2*(a^9*b - 2*a^7*b^3 + a^5*b^5)*d + ((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*d*cos(d*x + c)^2 - (a^10 - a^8*b^2 - a^6*b^4 + a^4*b^6)*d)*sin( d*x + c)), -1/2*((5*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^3 - (8*a^ 6*b - 17*a^4*b^3 + 9*a^2*b^5)*cos(d*x + c)*sin(d*x + c) + (4*a^5*b - 18*a^ 3*b^3 + 12*a*b^5 - 2*(2*a^5*b - 9*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^2 + (2*a ^6 - 7*a^4*b^2 - 3*a^2*b^4 + 6*b^6 - (2*a^4*b^2 - 9*a^2*b^4 + 6*b^6)*co...
\[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cot(d*x+c)**2/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**2/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^2/(a+b*sin(d*x+c))^3,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 = 0.17 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (2 \, a^{4} - 9 \, a^{2} b^{2} + 6 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b - 5 \, a^{2} b^{3}\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}} + \frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {6 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \] Input:
integrate(cot(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/2*(2*(2*a^4 - 9*a^2*b^2 + 6*b^4)*(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)))/((a^6 - a^4*b^2 )*sqrt(a^2 - b^2)) + 2*(5*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 6*a*b^4*tan(1/2 *d*x + 1/2*c)^3 + 4*a^4*b*tan(1/2*d*x + 1/2*c)^2 + 3*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 - 10*b^5*tan(1/2*d*x + 1/2*c)^2 + 11*a^3*b^2*tan(1/2*d*x + 1/2*c ) - 14*a*b^4*tan(1/2*d*x + 1/2*c) + 4*a^4*b - 5*a^2*b^3)/((a^6 - a^4*b^2)* (a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2) + 6*b*log(abs (tan(1/2*d*x + 1/2*c)))/a^4 - tan(1/2*d*x + 1/2*c)/a^3 - (6*b*tan(1/2*d*x + 1/2*c) - a)/(a^4*tan(1/2*d*x + 1/2*c)))/d
Time = 18.71 (sec) , antiderivative size = 1762, normalized size of antiderivative = 8.72 \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^2/(a + b*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)/(2*a^3*d) - (a^2 - (2*tan(c/2 + (d*x)/2)*(7*a*b^3 - 6*a ^3*b))/(a^2 - b^2) + (tan(c/2 + (d*x)/2)^4*(a^4 - 12*b^4 + 9*a^2*b^2))/(a^ 2 - b^2) + (2*tan(c/2 + (d*x)/2)^2*(a^4 - 16*b^4 + 12*a^2*b^2))/(a^2 - b^2 ) + (2*tan(c/2 + (d*x)/2)^3*(6*a^4*b - 10*b^5 + a^2*b^3))/(a*(a^2 - b^2))) /(d*(2*a^5*tan(c/2 + (d*x)/2)^5 + tan(c/2 + (d*x)/2)^3*(4*a^5 + 8*a^3*b^2) + 2*a^5*tan(c/2 + (d*x)/2) + 8*a^4*b*tan(c/2 + (d*x)/2)^2 + 8*a^4*b*tan(c /2 + (d*x)/2)^4)) - (3*b*log(tan(c/2 + (d*x)/2)))/(a^4*d) - (atan((((-(a + b)^3*(a - b)^3)^(1/2)*(a^4 + 3*b^4 - (9*a^2*b^2)/2)*((2*a^8 + 12*a^4*b^4 - 15*a^6*b^2)/(a^8 - a^6*b^2) + (tan(c/2 + (d*x)/2)*(10*a^8*b - 24*a^2*b^7 + 60*a^4*b^5 - 46*a^6*b^3))/(a^9 + a^5*b^4 - 2*a^7*b^2) + (((2*a^10*b - 2 *a^8*b^3)/(a^8 - a^6*b^2) - (tan(c/2 + (d*x)/2)*(6*a^12 - 8*a^6*b^6 + 22*a ^8*b^4 - 20*a^10*b^2))/(a^9 + a^5*b^4 - 2*a^7*b^2))*(-(a + b)^3*(a - b)^3) ^(1/2)*(a^4 + 3*b^4 - (9*a^2*b^2)/2))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8* b^2))*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2) + ((-(a + b)^3*(a - b)^ 3)^(1/2)*(a^4 + 3*b^4 - (9*a^2*b^2)/2)*((2*a^8 + 12*a^4*b^4 - 15*a^6*b^2)/ (a^8 - a^6*b^2) + (tan(c/2 + (d*x)/2)*(10*a^8*b - 24*a^2*b^7 + 60*a^4*b^5 - 46*a^6*b^3))/(a^9 + a^5*b^4 - 2*a^7*b^2) - (((2*a^10*b - 2*a^8*b^3)/(a^8 - a^6*b^2) - (tan(c/2 + (d*x)/2)*(6*a^12 - 8*a^6*b^6 + 22*a^8*b^4 - 20*a^ 10*b^2))/(a^9 + a^5*b^4 - 2*a^7*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(a^4 + 3*b^4 - (9*a^2*b^2)/2))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*1i)/(...
Time = 0.19 (sec) , antiderivative size = 1147, normalized size of antiderivative = 5.68 \[ \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^2/(a+b*sin(d*x+c))^3,x)
Output:
( - 16*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(c + d*x)**3*a**4*b**3 + 72*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**5 - 48*sqrt(a**2 - b**2)*a tan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*b**7 - 32* sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**5*b**2 + 144*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/s qrt(a**2 - b**2))*sin(c + d*x)**2*a**3*b**4 - 96*sqrt(a**2 - b**2)*atan((t an((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a*b**6 - 16*sqrt (a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x )*a**6*b + 72*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**4*b**3 - 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2 )*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**2*b**5 - 20*cos(c + d*x)*sin(c + d*x)**2*a**5*b**3 + 44*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 - 24*cos( c + d*x)*sin(c + d*x)**2*a*b**7 - 32*cos(c + d*x)*sin(c + d*x)*a**6*b**2 + 68*cos(c + d*x)*sin(c + d*x)*a**4*b**4 - 36*cos(c + d*x)*sin(c + d*x)*a** 2*b**6 - 8*cos(c + d*x)*a**7*b + 16*cos(c + d*x)*a**5*b**3 - 8*cos(c + d*x )*a**3*b**5 - 24*log(tan((c + d*x)/2))*sin(c + d*x)**3*a**4*b**4 + 48*log( tan((c + d*x)/2))*sin(c + d*x)**3*a**2*b**6 - 24*log(tan((c + d*x)/2))*sin (c + d*x)**3*b**8 - 48*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**5*b**3 + 9 6*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**3*b**5 - 48*log(tan((c + d*x...