Integrand size = 21, antiderivative size = 232 \[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d \sqrt {d \cot (e+f x)}}{f}-\frac {2 (d \cot (e+f x))^{5/2}}{5 d f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
-2/5*(d*cot(f*x+e))^(5/2)/d/f+1/2*d^(3/2)*arctan(1-2^(1/2)*(d*cot(f*x+e))^ (1/2)/d^(1/2))/f*2^(1/2)-1/2*d^(3/2)*arctan(1+2^(1/2)*(d*cot(f*x+e))^(1/2) /d^(1/2))/f*2^(1/2)+1/4*d^(3/2)*ln(d^(1/2)+cot(f*x+e)*d^(1/2)-2^(1/2)*(d*c ot(f*x+e))^(1/2))/f*2^(1/2)-1/4*d^(3/2)*ln(d^(1/2)+cot(f*x+e)*d^(1/2)+2^(1 /2)*(d*cot(f*x+e))^(1/2))/f*2^(1/2)+2*d*(d*cot(f*x+e))^(1/2)/f
Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.74 \[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=-\frac {(d \cot (e+f x))^{3/2} \left (-10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (e+f x)}\right )+10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (e+f x)}\right )-40 \sqrt {\cot (e+f x)}+8 \cot ^{\frac {5}{2}}(e+f x)-5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )+5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )\right )}{20 f \cot ^{\frac {3}{2}}(e+f x)} \]
-1/20*((d*Cot[e + f*x])^(3/2)*(-10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] - 40*Sqrt[Cot[ e + f*x]] + 8*Cot[e + f*x]^(5/2) - 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] + 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot [e + f*x]]))/(f*Cot[e + f*x]^(3/2))
Time = 0.59 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {2030, 3042, 3954, 3042, 3954, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle \frac {\int (d \cot (e+f x))^{7/2}dx}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}dx}{d^2}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-d^2 \int (d \cot (e+f x))^{3/2}dx-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-d^2 \int \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-d^2 \left (d^2 \left (-\int \frac {1}{\sqrt {d \cot (e+f x)}}dx\right )-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-d^2 \left (d^2 \left (-\int \frac {1}{\sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {-d^2 \left (\frac {d^3 \int \frac {1}{\sqrt {d \cot (e+f x)} \left (\cot ^2(e+f x) d^2+d^2\right )}d(d \cot (e+f x))}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \int \frac {1}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-d^2 \left (\frac {2 d^3 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 d \sqrt {d \cot (e+f x)}}{f}\right )-\frac {2 d (d \cot (e+f x))^{5/2}}{5 f}}{d^2}\) |
((-2*d*(d*Cot[e + f*x])^(5/2))/(5*f) - d^2*((-2*d*Sqrt[d*Cot[e + f*x]])/f + (2*d^3*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Cot[e + f*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*Sqrt[d]*Cot[e + f*x]]/(Sqrt[2]*Sqrt[d]))/(2*d) + (-1/2 *Log[d - Sqrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(Sqrt[2]*Sqrt[ d]) + Log[d + Sqrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(2*Sqrt[2 ]*Sqrt[d]))/(2*d)))/f))/d^2
3.3.6.3.1 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_)^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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Time = 0.14 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\left (\cot \left (f x +e \right ) d \right )^{\frac {5}{2}}}{5}-d^{2} \sqrt {\cot \left (f x +e \right ) d}+\frac {d^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\cot \left (f x +e \right ) d +\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}{\cot \left (f x +e \right ) d -\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{f d}\) | \(171\) |
| default | \(-\frac {2 \left (\frac {\left (\cot \left (f x +e \right ) d \right )^{\frac {5}{2}}}{5}-d^{2} \sqrt {\cot \left (f x +e \right ) d}+\frac {d^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\cot \left (f x +e \right ) d +\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}{\cot \left (f x +e \right ) d -\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{f d}\) | \(171\) |
-2/f/d*(1/5*(cot(f*x+e)*d)^(5/2)-d^2*(cot(f*x+e)*d)^(1/2)+1/8*d^2*(d^2)^(1 /4)*2^(1/2)*(ln((cot(f*x+e)*d+(d^2)^(1/4)*(cot(f*x+e)*d)^(1/2)*2^(1/2)+(d^ 2)^(1/2))/(cot(f*x+e)*d-(d^2)^(1/4)*(cot(f*x+e)*d)^(1/2)*2^(1/2)+(d^2)^(1/ 2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(cot(f*x+e)*d)^(1/2)+1)-2*arctan(-2^(1/2 )/(d^2)^(1/4)*(cot(f*x+e)*d)^(1/2)+1)))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.51 \[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=-\frac {5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} {\left (f \cos \left (2 \, f x + 2 \, e\right ) - f\right )} \log \left (d \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}} + \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + 5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} {\left (i \, f \cos \left (2 \, f x + 2 \, e\right ) - i \, f\right )} \log \left (d \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}} + i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + 5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} {\left (-i \, f \cos \left (2 \, f x + 2 \, e\right ) + i \, f\right )} \log \left (d \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}} - i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) - 5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} {\left (f \cos \left (2 \, f x + 2 \, e\right ) - f\right )} \log \left (d \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}} - \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) - 8 \, {\left (3 \, d \cos \left (2 \, f x + 2 \, e\right ) - 2 \, d\right )} \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}}}{10 \, {\left (f \cos \left (2 \, f x + 2 \, e\right ) - f\right )}} \]
-1/10*(5*(-d^6/f^4)^(1/4)*(f*cos(2*f*x + 2*e) - f)*log(d*sqrt((d*cos(2*f*x + 2*e) + d)/sin(2*f*x + 2*e)) + (-d^6/f^4)^(1/4)*f) + 5*(-d^6/f^4)^(1/4)* (I*f*cos(2*f*x + 2*e) - I*f)*log(d*sqrt((d*cos(2*f*x + 2*e) + d)/sin(2*f*x + 2*e)) + I*(-d^6/f^4)^(1/4)*f) + 5*(-d^6/f^4)^(1/4)*(-I*f*cos(2*f*x + 2* e) + I*f)*log(d*sqrt((d*cos(2*f*x + 2*e) + d)/sin(2*f*x + 2*e)) - I*(-d^6/ f^4)^(1/4)*f) - 5*(-d^6/f^4)^(1/4)*(f*cos(2*f*x + 2*e) - f)*log(d*sqrt((d* cos(2*f*x + 2*e) + d)/sin(2*f*x + 2*e)) - (-d^6/f^4)^(1/4)*f) - 8*(3*d*cos (2*f*x + 2*e) - 2*d)*sqrt((d*cos(2*f*x + 2*e) + d)/sin(2*f*x + 2*e)))/(f*c os(2*f*x + 2*e) - f)
\[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (e + f x \right )}\, dx \]
Time = 0.47 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.86 \[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=-\frac {10 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right ) + 10 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right ) + 5 \, \sqrt {2} d^{\frac {5}{2}} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right ) - 5 \, \sqrt {2} d^{\frac {5}{2}} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right ) - 40 \, d^{2} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + 8 \, \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}{20 \, d f} \]
-1/20*(10*sqrt(2)*d^(5/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d/t an(f*x + e)))/sqrt(d)) + 10*sqrt(2)*d^(5/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*s qrt(d) - 2*sqrt(d/tan(f*x + e)))/sqrt(d)) + 5*sqrt(2)*d^(5/2)*log(sqrt(2)* sqrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e)) - 5*sqrt(2)*d^(5/2)*log (-sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e)) - 40*d^2*sqrt (d/tan(f*x + e)) + 8*(d/tan(f*x + e))^(5/2))/(d*f)
\[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}} \cot \left (f x + e\right )^{2} \,d x } \]
Time = 3.65 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.39 \[ \int \cot ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx=\frac {2\,d\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}}{f}-\frac {2\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}\,1{}\mathrm {i}}{\sqrt {d}}\right )}{f} \]