\(\int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx\) [208]

Optimal result

Integrand size = 21, antiderivative size = 156 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}+\sqrt {d} \cot (e+f x)}\right )}{\sqrt {2} \sqrt {d} f}+\frac {2 d}{3 f (d \cot (e+f x))^{3/2}} \] Output:

-1/2*arctan(1-2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/d^(1/2)/f+1/2* 
arctan(1+2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/d^(1/2)/f+1/2*arcta 
nh(2^(1/2)*(d*cot(f*x+e))^(1/2)/(d^(1/2)+d^(1/2)*cot(f*x+e)))*2^(1/2)/d^(1 
/2)/f+2/3*d/f/(d*cot(f*x+e))^(3/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.54 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {d \left (-2+3 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{3/4}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{3/4}\right )}{3 f (d \cot (e+f x))^{3/2}} \] Input:

Integrate[Tan[e + f*x]^2/Sqrt[d*Cot[e + f*x]],x]
 

Output:

-1/3*(d*(-2 + 3*ArcTan[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(3/4) + 
3*ArcTanh[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(3/4)))/(f*(d*Cot[e + 
 f*x])^(3/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.34, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 2030, 3955, 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.

\(\displaystyle \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan \left (e+f x+\frac {\pi }{2}\right )^2 \sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 2030

\(\displaystyle d^2 \int \frac {1}{\left (-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 3955

\(\displaystyle d^2 \left (\frac {2}{3 d f (d \cot (e+f x))^{3/2}}-\frac {\int \frac {1}{\sqrt {d \cot (e+f x)}}dx}{d^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\frac {2}{3 d f (d \cot (e+f x))^{3/2}}-\frac {\int \frac {1}{\sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}}dx}{d^2}\right )\)

\(\Big \downarrow \) 3957

\(\displaystyle d^2 \left (\frac {\int \frac {1}{\sqrt {d \cot (e+f x)} \left (\cot ^2(e+f x) d^2+d^2\right )}d(d \cot (e+f x))}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle d^2 \left (\frac {2 \int \frac {1}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle d^2 \left (\frac {2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle d^2 \left (\frac {2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{d f}+\frac {2}{3 d f (d \cot (e+f x))^{3/2}}\right )\)

Input:

Int[Tan[e + f*x]^2/Sqrt[d*Cot[e + f*x]],x]
 

Output:

d^2*(2/(3*d*f*(d*Cot[e + f*x])^(3/2)) + (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)))/(d*f))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 266

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]
 

rule 755

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]]))
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 2030

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3955

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] 
)^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2   Int[(b*Tan[c + d*x])^(n + 2), x] 
, x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
 

rule 3957

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(497\) vs. \(2(122)=244\).

Time = 1.38 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.19

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 {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+\csc \left (f x +e \right )-\sin \left (f x +e \right )-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (-3 \cot \left (f x +e \right )-3 \csc \left (f x +e \right )\right )+\sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right ) \left (-6 \cot \left (f x +e \right )-6 \csc \left (f x +e \right )\right )+\sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \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 {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+\csc \left (f x +e \right )-\sin \left (f x +e \right )-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right ) \left (3 \cot \left (f x +e \right )+3 \csc \left (f x +e \right )\right )+\sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \arctan \left (\frac {-\sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right ) \left (6 \cot \left (f x +e \right )+6 \csc \left (f x +e \right )\right )-4 \sqrt {2}\, \tan \left (f x +e \right )\right ) \sqrt {2}}{12 f \sqrt {d \cot \left (f x +e \right )}}\)	\(498\)

Input:

int(tan(f*x+e)^2/(d*cot(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/12/f/(d*cot(f*x+e))^(1/2)*(ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e)-2*si 
n(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+csc(f*x+e)-sin( 
f*x+e)-2*cos(f*x+e)+2)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+ 
e))^2)^(1/2)*(-3*cot(f*x+e)-3*csc(f*x+e))+(-sin(f*x+e)*cos(f*x+e)/(1+cos(f 
*x+e))^2)^(1/2)*arctan((sin(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e) 
)^2)^(1/2)+cos(f*x+e)-1)/(cos(f*x+e)-1))*(-6*cot(f*x+e)-6*csc(f*x+e))+(-si 
n(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*ln(-(cot(f*x+e)*cos(f*x+e)-2*c 
ot(f*x+e)+2*sin(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+c 
sc(f*x+e)-sin(f*x+e)-2*cos(f*x+e)+2)/(cos(f*x+e)-1))*(3*cot(f*x+e)+3*csc(f 
*x+e))+(-sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*arctan((-sin(f*x+e) 
*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+cos(f*x+e)-1)/(cos(f*x+ 
e)-1))*(6*cot(f*x+e)+6*csc(f*x+e))-4*2^(1/2)*tan(f*x+e))*2^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.26 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {8 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )^{2} + 6 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{\sqrt {d}} + 1\right ) + 6 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{\sqrt {d}} - 1\right ) + 3 \, \sqrt {2} \sqrt {d} \log \left (\frac {\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{\sqrt {d}} + \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )}\right ) - 3 \, \sqrt {2} \sqrt {d} \log \left (-\frac {\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{\sqrt {d}} - \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )}\right )}{12 \, d f} \] Input:

integrate(tan(f*x+e)^2/(d*cot(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

1/12*(8*sqrt(d/tan(f*x + e))*tan(f*x + e)^2 + 6*sqrt(2)*sqrt(d)*arctan(sqr 
t(2)*sqrt(d/tan(f*x + e))/sqrt(d) + 1) + 6*sqrt(2)*sqrt(d)*arctan(sqrt(2)* 
sqrt(d/tan(f*x + e))/sqrt(d) - 1) + 3*sqrt(2)*sqrt(d)*log((sqrt(2)*sqrt(d/ 
tan(f*x + e))*tan(f*x + e)/sqrt(d) + tan(f*x + e) + 1)/tan(f*x + e)) - 3*s 
qrt(2)*sqrt(d)*log(-(sqrt(2)*sqrt(d/tan(f*x + e))*tan(f*x + e)/sqrt(d) - t 
an(f*x + e) - 1)/tan(f*x + e)))/(d*f)
 

Sympy [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\sqrt {d \cot {\left (e + f x \right )}}}\, dx \] Input:

integrate(tan(f*x+e)**2/(d*cot(f*x+e))**(1/2),x)
 

Output:

Integral(tan(e + f*x)**2/sqrt(d*cot(e + f*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.22 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {d^{3} {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{d^{2}} + \frac {8}{d^{2} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {3}{2}}}\right )}}{12 \, f} \] Input:

integrate(tan(f*x+e)^2/(d*cot(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

1/12*d^3*(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 
)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/d^(3/2) - sqrt(2)*log(-sqrt(2 
)*sqrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/d^(3/2))/d^2 + 8/(d^2 
*(d/tan(f*x + e))^(3/2)))/f
 

Giac [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\sqrt {d \cot \left (f x + e\right )}} \,d x } \] Input:

integrate(tan(f*x+e)^2/(d*cot(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(tan(f*x + e)^2/sqrt(d*cot(f*x + e)), x)
 

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.52 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2\,d}{3\,f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{\sqrt {d}\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{\sqrt {d}\,f} \] Input:

int(tan(e + f*x)^2/(d*cot(e + f*x))^(1/2),x)
 

Output:

(2*d)/(3*f*(d/tan(e + f*x))^(3/2)) - ((-1)^(1/4)*atan(((-1)^(1/4)*(d/tan(e 
 + f*x))^(1/2))/d^(1/2))*1i)/(d^(1/2)*f) - ((-1)^(1/4)*atanh(((-1)^(1/4)*( 
d/tan(e + f*x))^(1/2))/d^(1/2))*1i)/(d^(1/2)*f)
 

Reduce [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\cot \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}}{\cot \left (f x +e \right )}d x \right )}{d} \] Input:

int(tan(f*x+e)^2/(d*cot(f*x+e))^(1/2),x)
 

Output:

(sqrt(d)*int((sqrt(cot(e + f*x))*tan(e + f*x)**2)/cot(e + f*x),x))/d