Integrand size = 25, antiderivative size = 161 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )} \]
-1/8*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a^3/d-1/4*arctanh(1/2*(e ^(1/2)+cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))*e^(1/2)/a^3/d*2^( 1/2)+1/4*(e*cot(d*x+c))^(1/2)/a/d/(a+a*cot(d*x+c))^2+3/8*(e*cot(d*x+c))^(1 /2)/d/(a^3+a^3*cot(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.92 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=-\frac {e \left (-\frac {\arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^3 \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )}{4 a^3 \sqrt [4]{-e^2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{4 \sqrt {2} a^3 \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{4 \sqrt {2} a^3 \sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )}{4 a^3 \sqrt [4]{-e^2}}-\frac {-e \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-e \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right ) \cot (c+d x)+\sqrt {e} \sqrt {e \cot (c+d x)}}{2 a^3 \sqrt {e} (e+e \cot (c+d x))}+\frac {(e \cot (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},-\cot (c+d x)\right )}{3 a^3 e^2}-\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{8 \sqrt {2} a^3 \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{8 \sqrt {2} a^3 \sqrt {e}}\right )}{d} \]
-((e*(-1/2*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a^3*Sqrt[e]) - ArcTan[Sqr t[e*Cot[c + d*x]]/(-e^2)^(1/4)]/(4*a^3*(-e^2)^(1/4)) - ArcTan[1 - (Sqrt[2] *Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(4*Sqrt[2]*a^3*Sqrt[e]) + ArcTan[1 + (Sqrt [2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(4*Sqrt[2]*a^3*Sqrt[e]) + ArcTanh[Sqrt[ e*Cot[c + d*x]]/(-e^2)^(1/4)]/(4*a^3*(-e^2)^(1/4)) - (-(e*ArcTan[Sqrt[e*Co t[c + d*x]]/Sqrt[e]]) - e*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]*Cot[c + d*x ] + Sqrt[e]*Sqrt[e*Cot[c + d*x]])/(2*a^3*Sqrt[e]*(e + e*Cot[c + d*x])) + ( (e*Cot[c + d*x])^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, -Cot[c + d*x]])/(3*a ^3*e^2) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x] ]]/(8*Sqrt[2]*a^3*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]* Sqrt[e*Cot[c + d*x]]]/(8*Sqrt[2]*a^3*Sqrt[e])))/d)
Time = 1.28 (sec) , antiderivative size = 175, 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, 4051, 27, 3042, 4132, 25, 3042, 4137, 27, 3042, 4015, 221, 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.
| \(\displaystyle \int \frac {\sqrt {e \cot (c+d x)}}{(a \cot (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4051 |
| \(\displaystyle \frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}-\frac {\int -\frac {-3 a e \cot ^2(c+d x)+4 a e \cot (c+d x)+a e}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)^2}dx}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-3 a e \cot ^2(c+d x)+4 a e \cot (c+d x)+a e}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)^2}dx}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-3 a e \tan \left (c+d x+\frac {\pi }{2}\right )^2-4 a e \tan \left (c+d x+\frac {\pi }{2}\right )+a e}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}-\frac {\int -\frac {5 a^3 e^2-3 a^3 e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^3 e}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {5 a^3 e^2-3 a^3 e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {5 a^3 e^2-3 a^3 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4137 |
| \(\displaystyle \frac {\frac {a^3 e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {\int \frac {8 \left (a^4 e^2-a^4 e^2 \cot (c+d x)\right )}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^3 e^2 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {4 \int \frac {a^4 e^2-a^4 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{a^2}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 \int \frac {e^2 a^4+e^2 \tan \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {\frac {a^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {8 a^6 e^4 \int \frac {1}{2 a^8 e^4-\left (e^2 a^4+e^2 \cot (c+d x) a^4\right )^2 \tan (c+d x)}d\frac {e^2 a^4+e^2 \cot (c+d x) a^4}{\sqrt {e \cot (c+d x)}}}{d}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {a^3 e^2 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 \sqrt {2} a^2 e^{3/2} \text {arctanh}\left (\frac {a^4 e^2 \cot (c+d x)+a^4 e^2}{\sqrt {2} a^4 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {a^3 e^2 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^2 e^{3/2} \text {arctanh}\left (\frac {a^4 e^2 \cot (c+d x)+a^4 e^2}{\sqrt {2} a^4 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {a^2 e^2 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^2 e^{3/2} \text {arctanh}\left (\frac {a^4 e^2 \cot (c+d x)+a^4 e^2}{\sqrt {2} a^4 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {2 a^2 e \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}-\frac {4 \sqrt {2} a^2 e^{3/2} \text {arctanh}\left (\frac {a^4 e^2 \cot (c+d x)+a^4 e^2}{\sqrt {2} a^4 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 e^{3/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}-\frac {4 \sqrt {2} a^2 e^{3/2} \text {arctanh}\left (\frac {a^4 e^2 \cot (c+d x)+a^4 e^2}{\sqrt {2} a^4 e^{3/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}+\frac {3 \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\) |
Sqrt[e*Cot[c + d*x]]/(4*a*d*(a + a*Cot[c + d*x])^2) + (((2*a^2*e^(3/2)*Arc Tan[Cot[c + d*x]/Sqrt[e]])/d - (4*Sqrt[2]*a^2*e^(3/2)*ArcTanh[(a^4*e^2 + a ^4*e^2*Cot[c + d*x])/(Sqrt[2]*a^4*e^(3/2)*Sqrt[e*Cot[c + d*x]])])/d)/(2*a^ 3*e) + (3*Sqrt[e*Cot[c + d*x]])/(d*(a + a*Cot[c + d*x])))/(8*a^2)
3.1.37.3.1 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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2 )) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c *(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && Int egerQ[2*m]
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_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Sim p[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*C)*T an[e + f*x], x], x], x] + Simp[(A*b^2 + 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, 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. \(2(132)=264\).
Time = 0.05 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.17
| method | result | size |
| derivativedivides | \(-\frac {2 e^{4} \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{3}}-\frac {\frac {\frac {3 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {5 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}-\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{3}}\right )}{d \,a^{3}}\) | \(349\) |
| default | \(-\frac {2 e^{4} \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{3}}-\frac {\frac {\frac {3 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {5 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}-\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{3}}\right )}{d \,a^{3}}\) | \(349\) |
-2/d/a^3*e^4*(1/4/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* 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^3*((3/4*(e*cot (d*x+c))^(3/2)+5/4*e*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)+e)^2-1/4/e^(1/2)* arctan((e*cot(d*x+c))^(1/2)/e^(1/2))))
Time = 0.28 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.22 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=\left [\frac {4 \, {\left (\sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {-e} \arctan \left (\frac {{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + \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 (5 \, \cos \left (2 \, d x + 2 \, c\right ) - 3 \, \sin \left (2 \, d x + 2 \, c\right ) - 5\right )}}{16 \, {\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}, -\frac {2 \, \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 ) - 2 \, {\left (\sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {e} \log \left ({\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt {2}\right )} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (5 \, \cos \left (2 \, d x + 2 \, c\right ) - 3 \, \sin \left (2 \, d x + 2 \, c\right ) - 5\right )}}{16 \, {\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}\right ] \]
[1/16*(4*(sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*sqrt(-e)*arctan(1/2*(sqrt(2) *cos(2*d*x + 2*c) + sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*sqrt(-e)*sqrt((e*c os(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/(e*cos(2*d*x + 2*c) + e)) + 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))*(5*cos(2*d*x + 2*c) - 3*sin(2*d*x + 2*c) - 5))/(a^ 3*d*sin(2*d*x + 2*c) + a^3*d), -1/16*(2*sqrt(e)*(sin(2*d*x + 2*c) + 1)*arc tan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) - 2*(sqrt(2)* sin(2*d*x + 2*c) + sqrt(2))*sqrt(e)*log((sqrt(2)*cos(2*d*x + 2*c) - sqrt(2 )*sin(2*d*x + 2*c) - sqrt(2))*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2* d*x + 2*c)) + 2*e*sin(2*d*x + 2*c) + e) + sqrt((e*cos(2*d*x + 2*c) + e)/si n(2*d*x + 2*c))*(5*cos(2*d*x + 2*c) - 3*sin(2*d*x + 2*c) - 5))/(a^3*d*sin( 2*d*x + 2*c) + a^3*d)]
\[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=\frac {\int \frac {\sqrt {e \cot {\left (c + d x \right )}}}{\cot ^{3}{\left (c + d x \right )} + 3 \cot ^{2}{\left (c + d x \right )} + 3 \cot {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(sqrt(e*cot(c + d*x))/(cot(c + d*x)**3 + 3*cot(c + d*x)**2 + 3*cot (c + d*x) + 1), x)/a**3
Exception generated. \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=\int { \frac {\sqrt {e \cot \left (d x + c\right )}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Time = 13.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx=\frac {\frac {3\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{8}+\frac {5\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{8}}{d\,a^3\,e^2\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,d\,a^3\,e^2\,\mathrm {cot}\left (c+d\,x\right )+d\,a^3\,e^2}-\frac {\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d}-\frac {\sqrt {2}\,\sqrt {e}\,\mathrm {atanh}\left (\frac {9\,\sqrt {2}\,e^{17/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,\left (\frac {9\,e^9\,\mathrm {cot}\left (c+d\,x\right )}{32}+\frac {9\,e^9}{32}\right )}\right )}{4\,a^3\,d} \]
((3*e*(e*cot(c + d*x))^(3/2))/8 + (5*e^2*(e*cot(c + d*x))^(1/2))/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)) - (e^(1/2)*ata n((e*cot(c + d*x))^(1/2)/e^(1/2)))/(8*a^3*d) - (2^(1/2)*e^(1/2)*atanh((9*2 ^(1/2)*e^(17/2)*(e*cot(c + d*x))^(1/2))/(32*((9*e^9*cot(c + d*x))/32 + (9* e^9)/32))))/(4*a^3*d)