\(\int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx\) [209]

Optimal result

Integrand size = 19, antiderivative size = 154 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}+\sqrt {d} \cot (e+f x)}\right )}{\sqrt {2} \sqrt {d} f}+\frac {2}{f \sqrt {d \cot (e+f x)}} \] 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/f/(d*cot(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.51 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}}{f \sqrt {d \cot (e+f x)}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.48 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {3042, 25, 2030, 3955, 3042, 3957, 266, 826, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2030

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

\(\Big \downarrow \) 3955

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

d*(2/(d*f*Sqrt[d*Cot[e + f*x]]) + (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]]/(Sqr 
t[2]*Sqrt[d]))/2 + (Log[d - Sqrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x 
]^2]/(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))
 

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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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. \(858\) vs. \(2(121)=242\).

Time = 2.38 (sec) , antiderivative size = 859, normalized size of antiderivative = 5.58

method	result	size
default	\(\text {Expression too large to display}\)	\(859\)

Input:

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

Output:

-1/4/f*csc(f*x+e)*(ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(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)+csc(f*x+e)-sin(f*x+e)-2*co 
s(f*x+e)+2)/(cos(f*x+e)-1))*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1 
/2)*cos(f*x+e)-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))*(-2*sin(f*x+e)*cos(f*x+e)/(1+co 
s(f*x+e))^2)^(1/2)*cos(f*x+e)-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))*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f* 
x+e))^2)^(1/2)*cos(f*x+e)+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))*(-2*sin(f*x+e)*cos(f* 
x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(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)+csc(f 
*x+e)-sin(f*x+e)-2*cos(f*x+e)+2)/(cos(f*x+e)-1))*(-2*sin(f*x+e)*cos(f*x+e) 
/(1+cos(f*x+e))^2)^(1/2)-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))*(-2*sin(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*cot(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)+csc(f*x+e)-sin(f 
*x+e)-2*cos(f*x+e)+2)/(cos(f*x+e)-1))*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x 
+e))^2)^(1/2)+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))*(-2*sin(f*x+e)*cos(f*x+e)/(1+c...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.25 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{\sqrt {d}} + 1\right ) + 2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{\sqrt {d}} - 1\right ) - \sqrt {2} \sqrt {d} \log \left (\frac {\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{\sqrt {d}} + \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )}\right ) + \sqrt {2} \sqrt {d} \log \left (-\frac {\frac {\sqrt {2} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{\sqrt {d}} - \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )}\right ) + 8 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{4 \, d f} \] Input:

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

Output:

1/4*(2*sqrt(2)*sqrt(d)*arctan(sqrt(2)*sqrt(d/tan(f*x + e))/sqrt(d) + 1) + 
2*sqrt(2)*sqrt(d)*arctan(sqrt(2)*sqrt(d/tan(f*x + e))/sqrt(d) - 1) - 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)) + 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)) + 8*sqrt(d/tan 
(f*x + e))*tan(f*x + e))/(d*f)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.23 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {d^{2} {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}}{d^{2}} + \frac {8}{d^{2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}\right )}}{4 \, f} \] Input:

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

Output:

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))/sqrt(d) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) 
- 2*sqrt(d/tan(f*x + e)))/sqrt(d))/sqrt(d) - sqrt(2)*log(sqrt(2)*sqrt(d)*s 
qrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d) + sqrt(2)*log(-sqrt(2)*s 
qrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d))/d^2 + 8/(d^2*sq 
rt(d/tan(f*x + e))))/f
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.51 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2}{f\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{\sqrt {d}\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{\sqrt {d}\,f} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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