\(\int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx\) [203]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}+\sqrt {d} \cot (e+f x)}\right )}{\sqrt {2} f} \] Output:

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.53 \[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=\frac {d \left (-\arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right )+\text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right )\right ) \sqrt [4]{-\cot (e+f x)} \sqrt {d \cot (e+f x)}}{f \cot ^{\frac {3}{4}}(e+f x)} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.29, 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, 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.

\(\displaystyle \int \tan (e+f x) (d \cot (e+f x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2030

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-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 + (Log[d - S 
qrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]) - Lo 
g[d + Sqrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d 
]))/2))/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 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 [A] (verified)

Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d \cot \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \cot \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \cot \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \cot \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \cot \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \cot \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f \left (d^{2}\right )^{\frac {1}{4}}}\)	\(138\)

Input:

int((d*cot(f*x+e))^(3/2)*tan(f*x+e),x,method=_RETURNVERBOSE)
 

Output:

-1/4*d^2/f/(d^2)^(1/4)*2^(1/2)*(ln((d*cot(f*x+e)-(d^2)^(1/4)*(d*cot(f*x+e) 
)^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*cot(f*x+e)+(d^2)^(1/4)*(d*cot(f*x+e))^(1/2 
)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*cot(f*x+e))^(1/2)+ 
1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*cot(f*x+e))^(1/2)+1))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.36 \[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=-\frac {2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d}{d}\right ) + 2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} - d}{d}\right ) - \sqrt {2} d^{\frac {3}{2}} \log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right ) + d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )}\right ) + \sqrt {2} d^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right ) - d \tan \left (f x + e\right ) - d}{\tan \left (f x + e\right )}\right )}{4 \, f} \] Input:

integrate((d*cot(f*x+e))^(3/2)*tan(f*x+e),x, algorithm="fricas")
 

Output:

-1/4*(2*sqrt(2)*d^(3/2)*arctan((sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e)) + d)/ 
d) + 2*sqrt(2)*d^(3/2)*arctan((sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e)) - d)/d 
) - sqrt(2)*d^(3/2)*log((sqrt(2)*sqrt(d)*sqrt(d/tan(f*x + e))*tan(f*x + e) 
 + d*tan(f*x + e) + d)/tan(f*x + e)) + sqrt(2)*d^(3/2)*log(-(sqrt(2)*sqrt( 
d)*sqrt(d/tan(f*x + e))*tan(f*x + e) - d*tan(f*x + e) - d)/tan(f*x + e)))/ 
f
 

Sympy [F]

\[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\, dx \] Input:

integrate((d*cot(f*x+e))**(3/2)*tan(f*x+e),x)
 

Output:

Integral((d*cot(e + f*x))**(3/2)*tan(e + f*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=-\frac {d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{4 \, f} \] Input:

integrate((d*cot(f*x+e))^(3/2)*tan(f*x+e),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))/f
 

Giac [F]

\[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}} \tan \left (f x + e\right ) \,d x } \] Input:

integrate((d*cot(f*x+e))^(3/2)*tan(f*x+e),x, algorithm="giac")
 

Output:

integrate((d*cot(f*x + e))^(3/2)*tan(f*x + e), x)
 

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.40 \[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=-\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\left (\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )-\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\right )}{f} \] Input:

int(tan(e + f*x)*(d*cot(e + f*x))^(3/2),x)
 

Output:

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

Reduce [F]

\[ \int (d \cot (e+f x))^{3/2} \tan (e+f x) \, dx=\sqrt {d}\, \left (\int \sqrt {\cot \left (f x +e \right )}\, \cot \left (f x +e \right ) \tan \left (f x +e \right )d x \right ) d \] Input:

int((d*cot(f*x+e))^(3/2)*tan(f*x+e),x)
 

Output:

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