Integrand size = 25, antiderivative size = 165 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=-\frac {11 \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d \sqrt {e}}-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2} \]
-11/8*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a^3/d/e^(1/2)-1/4*arctan(1/2*(e ^(1/2)-cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))/a^3/d*2^(1/2)/e^( 1/2)-7/8*(e*cot(d*x+c))^(1/2)/a^3/d/e/(1+cot(d*x+c))-1/4*(e*cot(d*x+c))^(1 /2)/a/d/e/(a+a*cot(d*x+c))^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.39 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=-\frac {16 e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+4 \left (-e^2\right )^{3/4} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )+2 \sqrt {2} e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-2 \sqrt {2} e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-4 \left (-e^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )+\frac {8 e \sqrt {e \cot (c+d x)}}{1+\cot (c+d x)}+16 e \sqrt {e \cot (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},-\cot (c+d x)\right )+\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )-\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{16 a^3 d e^2} \]
-1/16*(16*e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]] + 4*(-e^2)^(3/4)*Ar cTan[Sqrt[e*Cot[c + d*x]]/(-e^2)^(1/4)] + 2*Sqrt[2]*e^(3/2)*ArcTan[1 - (Sq rt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - 2*Sqrt[2]*e^(3/2)*ArcTan[1 + (Sqrt[ 2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - 4*(-e^2)^(3/4)*ArcTanh[Sqrt[e*Cot[c + d*x]]/(-e^2)^(1/4)] + (8*e*Sqrt[e*Cot[c + d*x]])/(1 + Cot[c + d*x]) + 16*e *Sqrt[e*Cot[c + d*x]]*Hypergeometric2F1[1/2, 3, 3/2, -Cot[c + d*x]] + Sqrt [2]*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d* x]]] - Sqrt[2]*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e *Cot[c + d*x]]])/(a^3*d*e^2)
Time = 1.34 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4052, 27, 3042, 4132, 25, 3042, 4136, 27, 3042, 4015, 218, 4117, 27, 73, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a \cot (c+d x)+a)^3 \sqrt {e \cot (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3 \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -\frac {\int -\frac {3 e \cot ^2(c+d x) a^2+7 e a^2-4 e \cot (c+d x) a^2}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)^2}dx}{4 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 e \cot ^2(c+d x) a^2+7 e a^2-4 e \cot (c+d x) a^2}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)^2}dx}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 e \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^2+7 e a^2+4 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int -\frac {7 e^2 a^4+7 e^2 \cot ^2(c+d x) a^4-8 e^2 \cot (c+d x) a^4}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {7 e^2 a^4+7 e^2 \cot ^2(c+d x) a^4-8 e^2 \cot (c+d x) a^4}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {7 e^2 a^4+7 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^4+8 e^2 \tan \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {11 a^4 e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {\int -\frac {8 \left (e^2 a^5+e^2 \cot (c+d x) a^5\right )}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {11 a^4 e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx-\frac {4 \int \frac {e^2 a^5+e^2 \cot (c+d x) a^5}{\sqrt {e \cot (c+d x)}}dx}{a^2}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {11 a^4 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 \int \frac {a^5 e^2-a^5 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {\frac {11 a^4 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {8 a^8 e^4 \int \frac {1}{-2 e^4 a^{10}-\left (a^5 e^2-a^5 e^2 \cot (c+d x)\right )^2 \tan (c+d x)}d\frac {a^5 e^2-a^5 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}}{d}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {11 a^4 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 \sqrt {2} a^3 e^{3/2} \arctan \left (\frac {a^5 e^2-a^5 e^2 \cot (c+d x)}{\sqrt {2} a^5 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {11 a^4 e^2 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^3 e^{3/2} \arctan \left (\frac {a^5 e^2-a^5 e^2 \cot (c+d x)}{\sqrt {2} a^5 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {11 a^3 e^2 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^3 e^{3/2} \arctan \left (\frac {a^5 e^2-a^5 e^2 \cot (c+d x)}{\sqrt {2} a^5 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {22 a^3 e \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}-\frac {4 \sqrt {2} a^3 e^{3/2} \arctan \left (\frac {a^5 e^2-a^5 e^2 \cot (c+d x)}{\sqrt {2} a^5 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {22 a^3 e^{3/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}-\frac {4 \sqrt {2} a^3 e^{3/2} \arctan \left (\frac {a^5 e^2-a^5 e^2 \cot (c+d x)}{\sqrt {2} a^5 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {7 \sqrt {e \cot (c+d x)}}{d (\cot (c+d x)+1)}}{8 a^3 e}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2}\) |
-1/4*Sqrt[e*Cot[c + d*x]]/(a*d*e*(a + a*Cot[c + d*x])^2) + (((22*a^3*e^(3/ 2)*ArcTan[Cot[c + d*x]/Sqrt[e]])/d - (4*Sqrt[2]*a^3*e^(3/2)*ArcTan[(a^5*e^ 2 - a^5*e^2*Cot[c + d*x])/(Sqrt[2]*a^5*e^(3/2)*Sqrt[e*Cot[c + d*x]])])/d)/ (2*a^3*e) - (7*Sqrt[e*Cot[c + d*x]])/(d*(1 + Cot[c + d*x])))/(8*a^3*e)
3.1.38.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[-2*(d^2/f) Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[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] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[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*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*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] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs. \(2(136)=272\).
Time = 0.06 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.12
| method | result | size |
| derivativedivides | \(-\frac {2 e^{4} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{4}}+\frac {\frac {\frac {7 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {9 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {11 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{4}}\right )}{d \,a^{3}}\) | \(349\) |
| default | \(-\frac {2 e^{4} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{4}}+\frac {\frac {\frac {7 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {9 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {11 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{4}}\right )}{d \,a^{3}}\) | \(349\) |
-2/d/a^3*e^4*(1/4/e^4*(-1/8/e*(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^ (1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)* (e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e *cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1 ))-1/8/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1 /2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^ (1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2 *arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))+1/4/e^4*((7/4*(e*co t(d*x+c))^(3/2)+9/4*e*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)+e)^2+11/4/e^(1/2 )*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))))
Time = 0.29 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.05 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=\left [-\frac {2 \, \sqrt {2} \sqrt {-e} {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (-\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 11 \, \sqrt {-e} {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (9 \, \cos \left (2 \, d x + 2 \, c\right ) - 7 \, \sin \left (2 \, d x + 2 \, c\right ) - 9\right )}}{16 \, {\left (a^{3} d e \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d e\right )}}, -\frac {4 \, \sqrt {2} \sqrt {e} {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + 22 \, \sqrt {e} {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) - \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (9 \, \cos \left (2 \, d x + 2 \, c\right ) - 7 \, \sin \left (2 \, d x + 2 \, c\right ) - 9\right )}}{16 \, {\left (a^{3} d e \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d e\right )}}\right ] \]
[-1/16*(2*sqrt(2)*sqrt(-e)*(sin(2*d*x + 2*c) + 1)*log(-sqrt(2)*sqrt(-e)*sq rt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) + sin(2*d* x + 2*c) - 1) - 2*e*sin(2*d*x + 2*c) + e) + 11*sqrt(-e)*(sin(2*d*x + 2*c) + 1)*log((e*cos(2*d*x + 2*c) - e*sin(2*d*x + 2*c) + 2*sqrt(-e)*sqrt((e*cos (2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) + e)/(cos(2*d*x + 2* c) + sin(2*d*x + 2*c) + 1)) - sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2* c))*(9*cos(2*d*x + 2*c) - 7*sin(2*d*x + 2*c) - 9))/(a^3*d*e*sin(2*d*x + 2* c) + a^3*d*e), -1/16*(4*sqrt(2)*sqrt(e)*(sin(2*d*x + 2*c) + 1)*arctan(-1/2 *sqrt(2)*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(2*d* x + 2*c) - sin(2*d*x + 2*c) + 1)/(e*cos(2*d*x + 2*c) + e)) + 22*sqrt(e)*(s in(2*d*x + 2*c) + 1)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c) )/sqrt(e)) - sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(9*cos(2*d*x + 2*c) - 7*sin(2*d*x + 2*c) - 9))/(a^3*d*e*sin(2*d*x + 2*c) + a^3*d*e)]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=\frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )} + 3 \sqrt {e \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )} + 3 \sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a^{3}} \]
Integral(1/(sqrt(e*cot(c + d*x))*cot(c + d*x)**3 + 3*sqrt(e*cot(c + d*x))* cot(c + d*x)**2 + 3*sqrt(e*cot(c + d*x))*cot(c + d*x) + sqrt(e*cot(c + d*x ))), x)/a**3
Exception generated. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=\int { \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \sqrt {e \cot \left (d x + c\right )}} \,d x } \]
Time = 13.41 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx=\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{8\,a^3\,d\,\sqrt {e}}-\frac {11\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d\,\sqrt {e}}-\frac {\frac {9\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{8}+\frac {7\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{8}}{d\,a^3\,e^2\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,d\,a^3\,e^2\,\mathrm {cot}\left (c+d\,x\right )+d\,a^3\,e^2} \]
(2^(1/2)*(2*atan((2^(1/2)*(e*cot(c + d*x))^(1/2))/(2*e^(1/2))) + 2*atan((2 ^(1/2)*(e*cot(c + d*x))^(1/2))/(2*e^(1/2)) + (2^(1/2)*(e*cot(c + d*x))^(3/ 2))/(2*e^(3/2)))))/(8*a^3*d*e^(1/2)) - (11*atan((e*cot(c + d*x))^(1/2)/e^( 1/2)))/(8*a^3*d*e^(1/2)) - ((9*e*(e*cot(c + d*x))^(1/2))/8 + (7*(e*cot(c + d*x))^(3/2))/8)/(a^3*d*e^2 + a^3*d*e^2*cot(c + d*x)^2 + 2*a^3*d*e^2*cot(c + d*x))