Integrand size = 25, antiderivative size = 111 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\frac {\arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{3/2}}-\frac {\arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{3/2}}+\frac {2}{a d e \sqrt {e \cot (c+d x)}} \]
arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a/d/e^(3/2)-1/2*arctan(1/2*(e^(1/2)-c ot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))/a/d/e^(3/2)*2^(1/2)+2/a/d /e/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.06 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\frac {8 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\cot (c+d x)\right )+8 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\sqrt {2} \sqrt {e \cot (c+d x)} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )\right )}{8 a d e^{3/2} \sqrt {e \cot (c+d x)}} \]
(8*Sqrt[e]*Hypergeometric2F1[-1/2, 1, 1/2, -Cot[c + d*x]] + 8*Sqrt[e]*Hype rgeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*Sqrt[e*Cot[c + d*x] ]*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] + 2*ArcTan[1 + (S qrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]] + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sq rt[2]*Sqrt[e*Cot[c + d*x]]]))/(8*a*d*e^(3/2)*Sqrt[e*Cot[c + d*x]])
Time = 0.92 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4052, 27, 3042, 4136, 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.
| \(\displaystyle \int \frac {1}{(a \cot (c+d x)+a) (e \cot (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {2 \int -\frac {a \cot ^2(c+d x) e^2+a e^2+a \cot (c+d x) e^2}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}+\frac {2}{a d e \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a \cot ^2(c+d x) e^2+a e^2+a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^2+a e^2-a \tan \left (c+d x+\frac {\pi }{2}\right ) e^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {\int \frac {a^2 e^2+a^2 \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}+\frac {1}{2} a e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {\int \frac {a^2 e^2-a^2 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}+\frac {1}{2} a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {1}{2} a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {a^2 e^4 \int \frac {1}{-2 a^4 e^4-\left (a^2 e^2-a^2 e^2 \cot (c+d x)\right )^2 \tan (c+d x)}d\frac {a^2 e^2-a^2 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}}{d}}{a e^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {1}{2} a e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {e^{3/2} \arctan \left (\frac {a^2 e^2-a^2 e^2 \cot (c+d x)}{\sqrt {2} a^2 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}}{a e^3}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {a e^2 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{2 d}+\frac {e^{3/2} \arctan \left (\frac {a^2 e^2-a^2 e^2 \cot (c+d x)}{\sqrt {2} a^2 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}}{a e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {e^2 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{2 d}+\frac {e^{3/2} \arctan \left (\frac {a^2 e^2-a^2 e^2 \cot (c+d x)}{\sqrt {2} a^2 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}}{a e^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {e^{3/2} \arctan \left (\frac {a^2 e^2-a^2 e^2 \cot (c+d x)}{\sqrt {2} a^2 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}-\frac {e \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}}{a e^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\frac {e^{3/2} \arctan \left (\frac {a^2 e^2-a^2 e^2 \cot (c+d x)}{\sqrt {2} a^2 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}+\frac {e^{3/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}}{a e^3}\) |
-(((e^(3/2)*ArcTan[Cot[c + d*x]/Sqrt[e]])/d + (e^(3/2)*ArcTan[(a^2*e^2 - a ^2*e^2*Cot[c + d*x])/(Sqrt[2]*a^2*e^(3/2)*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2] *d))/(a*e^3)) + 2/(a*d*e*Sqrt[e*Cot[c + d*x]])
3.1.27.3.1 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[(((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. \(318\) vs. \(2(92)=184\).
Time = 0.04 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.87
| method | result | size |
| derivativedivides | \(-\frac {2 e^{2} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{2 e^{3}}-\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {7}{2}}}-\frac {1}{e^{3} \sqrt {e \cot \left (d x +c \right )}}\right )}{d a}\) | \(319\) |
| default | \(-\frac {2 e^{2} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{2 e^{3}}-\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 e^{\frac {7}{2}}}-\frac {1}{e^{3} \sqrt {e \cot \left (d x +c \right )}}\right )}{d a}\) | \(319\) |
-2/d/a*e^2*(1/2/e^3*(-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*c ot(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*a rctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/2/e^(7/2)*arctan((e *cot(d*x+c))^(1/2)/e^(1/2))-1/e^3/(e*cot(d*x+c))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (93) = 186\).
Time = 0.28 (sec) , antiderivative size = 472, normalized size of antiderivative = 4.25 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\left [-\frac {\sqrt {2} \sqrt {-e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (-\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 2 \, \sqrt {-e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - 8 \, \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{4 \, {\left (a d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a d e^{2}\right )}}, -\frac {\sqrt {2} \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - 2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) - 4 \, \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{2 \, {\left (a d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a d e^{2}\right )}}\right ] \]
[-1/4*(sqrt(2)*sqrt(-e)*(cos(2*d*x + 2*c) + 1)*log(-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) - 2*e*sin(2*d*x + 2*c) + e) + 2*sqrt(-e)*(cos(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)) - 8*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c) )*sin(2*d*x + 2*c))/(a*d*e^2*cos(2*d*x + 2*c) + a*d*e^2), -1/2*(sqrt(2)*sq rt(e)*(cos(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)) - 2*sqrt(e)*(cos(2*d*x + 2*c) + 1)*arctan(sqrt((e *cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) - 4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c))/(a*d*e^2*cos(2*d*x + 2*c) + a*d*e^2)]
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a} \]
Exception generated. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\int { \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 12.67 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx=\frac {2}{a\,d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{a\,d\,e^{3/2}}+\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{4\,a\,d\,e^{3/2}} \]