Integrand size = 19, antiderivative size = 192 \[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
1/2*arctan(1-2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/f*2^(1/2)-1/2*a rctan(1+2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/f*2^(1/2)+1/4*ln(d^( 1/2)+cot(f*x+e)*d^(1/2)-2^(1/2)*(d*cot(f*x+e))^(1/2))*d^(1/2)/f*2^(1/2)-1/ 4*ln(d^(1/2)+cot(f*x+e)*d^(1/2)+2^(1/2)*(d*cot(f*x+e))^(1/2))*d^(1/2)/f*2^ (1/2)
Time = 0.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.69 \[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=\frac {d \sqrt {\cot (e+f x)} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (e+f x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (e+f x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )\right )}{2 \sqrt {2} f \sqrt {d \cot (e+f x)}} \]
(d*Sqrt[Cot[e + f*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[e + f*x]]] - 2*ArcTan [1 + Sqrt[2]*Sqrt[Cot[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[e + f*x]] + Co t[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[e + f*x]] + Cot[e + f*x]]))/(2*Sqrt [2]*f*Sqrt[d*Cot[e + f*x]])
Time = 0.43 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.95, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {3042, 25, 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.
| \(\displaystyle \int \tan (e+f x) \sqrt {d \cot (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}}{\tan \left (e+f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sqrt {-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )}}{\tan \left (\frac {1}{2} (2 e+\pi )+f x\right )}dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle d \int \frac {1}{\sqrt {-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )}}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {d^2 \int \frac {1}{\sqrt {d \cot (e+f x)} \left (\cot ^2(e+f x) d^2+d^2\right )}d(d \cot (e+f x))}{f}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {2 d^2 \int \frac {1}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{f}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\) |
(-2*d^2*((-(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
3.2.94.3.1 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 = 14.92 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.33
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {d \left (\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}-\sin \left (f x +e \right )\right )}{1-\cos \left (f x +e \right )}}\, \left (\ln \left (\frac {\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+2-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right )+2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+1-\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right )-\ln \left (-\frac {-\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2+2 \cos \left (f x +e \right )+\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right )+2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-1+\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right )\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ) \sqrt {2}}{4 f \sqrt {\csc \left (f x +e \right ) \left (\left (\csc ^{2}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2}-1\right ) \left (1-\cos \left (f x +e \right )\right )}}\) | \(448\) |
-1/4/f*(-d/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2-sin(f*x+e)))^(1/2)* (ln(1/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2+2*sin(f*x+e)*(csc(f*x+e) ^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)+2-2*cos(f*x+e)-sin(f*x+e) ))+2*arctan(1/(1-cos(f*x+e))*(sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-cs c(f*x+e)+cot(f*x+e))^(1/2)+1-cos(f*x+e)))-ln(-1/(1-cos(f*x+e))*(-csc(f*x+e )*(1-cos(f*x+e))^2+2*sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+ cot(f*x+e))^(1/2)-2+2*cos(f*x+e)+sin(f*x+e)))+2*arctan(1/(1-cos(f*x+e))*(s in(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-1+co s(f*x+e))))/(csc(f*x+e)*(csc(f*x+e)^2*(1-cos(f*x+e))^2-1)*(1-cos(f*x+e)))^ (1/2)*(csc(f*x+e)-cot(f*x+e))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.81 \[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=-\frac {1}{2} \, \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - \frac {1}{2} i \, \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + \frac {1}{2} i \, \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + \frac {1}{2} \, \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) \]
-1/2*(-d^2/f^4)^(1/4)*log(f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e))) - 1/2 *I*(-d^2/f^4)^(1/4)*log(I*f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e))) + 1/2 *I*(-d^2/f^4)^(1/4)*log(-I*f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e))) + 1/ 2*(-d^2/f^4)^(1/4)*log(-f*(-d^2/f^4)^(1/4) + sqrt(d/tan(f*x + e)))
\[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=\int \sqrt {d \cot {\left (e + f x \right )}} \tan {\left (e + f x \right )}\, dx \]
Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=-\frac {d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{4 \, f} \]
-1/4*d^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) + 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
\[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=\int { \sqrt {d \cot \left (f x + e\right )} \tan \left (f x + e\right ) \,d x } \]
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.32 \[ \int \sqrt {d \cot (e+f x)} \tan (e+f x) \, dx=\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f}+\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f} \]