Integrand size = 21, antiderivative size = 192 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
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*cot(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)
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.70 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {d^2 \sqrt {\cot (e+f x)} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (e+f x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (e+f x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )\right )}{2 \sqrt {2} f \sqrt {d \cot (e+f x)}} \]
(d^2*Sqrt[Cot[e + f*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*ArcT an[1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(2*Sq rt[2]*f*Sqrt[d*Cot[e + f*x]])
Time = 0.44 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 2030, 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.
| \(\displaystyle \int \tan ^2(e+f x) (d \cot (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\tan \left (e+f x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle d^2 \int \frac {1}{\sqrt {-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )}}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {d^3 \int \frac {1}{\sqrt {d \cot (e+f x)} \left (\cot ^2(e+f x) d^2+d^2\right )}d(d \cot (e+f x))}{f}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {2 d^3 \int \frac {1}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{f}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 d^3 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
(-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
3.3.2.3.1 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/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]
Leaf count of result is larger than twice the leaf count of optimal. \(447\) vs. \(2(145)=290\).
Time = 2.05 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.33
| method | result | size |
| default | \(\frac {\left (\ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )-\ln \left (\frac {2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-\cot \left (f x +e \right ) \cos \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )+2 \cot \left (f x +e \right )-\csc \left (f x +e \right )-2}{\cos \left (f x +e \right )-1}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )\right ) d \sqrt {\cot \left (f x +e \right ) d}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ) \sqrt {2}}{4 f \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(448\) |
1/4/f*(ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-cot(f*x+e)^3 +3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(f*x+ e)-csc(f*x+e))^(1/2)-2*cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1)) -2*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x +e)-cos(f*x+e)+1)/(cos(f*x+e)-1))-ln((2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f* x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(f*x+e)-csc(f* x+e))^(1/2)-cot(f*x+e)*cos(f*x+e)+sin(f*x+e)+2*cos(f*x+e)+2*cot(f*x+e)-csc (f*x+e)-2)/(cos(f*x+e)-1))-2*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos( f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)-1)))*d*(cot(f*x+e) *d)^(1/2)/(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)-csc( f*x+e))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.85 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=-\frac {1}{2} \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} + \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) - \frac {1}{2} i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} + i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + \frac {1}{2} i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} - i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + \frac {1}{2} \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} - \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) \]
-1/2*(-d^6/f^4)^(1/4)*log(d*sqrt(d/tan(f*x + e)) + (-d^6/f^4)^(1/4)*f) - 1 /2*I*(-d^6/f^4)^(1/4)*log(d*sqrt(d/tan(f*x + e)) + I*(-d^6/f^4)^(1/4)*f) + 1/2*I*(-d^6/f^4)^(1/4)*log(d*sqrt(d/tan(f*x + e)) - I*(-d^6/f^4)^(1/4)*f) + 1/2*(-d^6/f^4)^(1/4)*log(d*sqrt(d/tan(f*x + e)) - (-d^6/f^4)^(1/4)*f)
\[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx \]
Time = 0.34 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=-\frac {d^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{4 \, f} \]
-1/4*d^3*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d/tan(f*x + e)))/sqrt(d))/d^(3/2) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d/tan(f*x + e)))/sqrt(d))/d^(3/2) + sqrt(2)*log(sqrt(2)*sqrt(d)*s qrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/d^(3/2) - sqrt(2)*log(-sqrt(2)*s qrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/d^(3/2))/f
\[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}} \tan \left (f x + e\right )^{2} \,d x } \]
Time = 3.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.32 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f} \]