3.2.91 \(\int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx\) [191]

3.2.91.1 Optimal result

Integrand size = 21, antiderivative size = 232 \[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^3}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d}{f \sqrt {d \cot (e+f x)}}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]

output

2/5*d^3/f/(d*cot(f*x+e))^(5/2)+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*arctan(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)-2*d/f/(d*cot(f*x+e))^(1/2)
 

3.2.91.2 Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=-\frac {\sqrt {d \cot (e+f x)} \left (-2+10 \cot ^2(e+f x)+5 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot (e+f x)} \cot ^{\frac {9}{4}}(e+f x)+5 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{5/4}\right ) \tan ^3(e+f x)}{5 f} \]

input

Integrate[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^4,x]
 

output

-1/5*(Sqrt[d*Cot[e + f*x]]*(-2 + 10*Cot[e + f*x]^2 + 5*ArcTan[(-Cot[e + f* 
x]^2)^(1/4)]*(-Cot[e + f*x])^(1/4)*Cot[e + f*x]^(9/4) + 5*ArcTanh[(-Cot[e 
+ f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(5/4))*Tan[e + f*x]^3)/f
 

3.2.91.3 Rubi [A] (warning: unable to verify)

Time = 0.61 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.99, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 2030, 3955, 3042, 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.

\(\displaystyle \int \tan ^4(e+f x) \sqrt {d \cot (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 2030

\(\displaystyle d^4 \int \frac {1}{\left (-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{7/2}}dx\)

\(\Big \downarrow \) 3955

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\int \frac {1}{(d \cot (e+f x))^{3/2}}dx}{d^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\int \frac {1}{\left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{d^2}\right )\)

\(\Big \downarrow \) 3955

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {d \cot (e+f x)}dx}{d^2}}{d^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{d^2}}{d^2}\right )\)

\(\Big \downarrow \) 3957

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {\int \frac {\sqrt {d \cot (e+f x)}}{\cot ^2(e+f x) d^2+d^2}d(d \cot (e+f x))}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \int \frac {d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 826

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle d^4 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\)

input

Int[Sqrt[d*Cot[e + f*x]]*Tan[e + f*x]^4,x]
 

output

d^4*(2/(5*d*f*(d*Cot[e + f*x])^(5/2)) - (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]]/(Sqrt[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 + S 
qrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]))/2)) 
/(d*f))/d^2)
 

3.2.91.3.1 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]
 

3.2.91.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(706\) vs. \(2(179)=358\).

Time = 17.60 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.05

method	result	size
default	\(-\frac {\csc \left (f x +e \right ) \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 (1-\cos \left (f x +e \right )\right ) \left (-40 \left (\csc ^{7}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{7}+5 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}-10 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}-5 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}-10 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}+112 \left (\csc ^{5}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{5}-40 \left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}\right ) \sqrt {2}}{20 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 )}\, {\left (\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 )\right )}^{\frac {5}{2}}}\)	\(707\)

input

int((cot(f*x+e)*d)^(1/2)*tan(f*x+e)^4,x,method=_RETURNVERBOSE)
 

output

-1/20/f*csc(f*x+e)*(-d/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2-sin(f*x 
+e)))^(1/2)*(1-cos(f*x+e))*(-40*csc(f*x+e)^7*(1-cos(f*x+e))^7+5*ln(1/(1-co 
s(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)))*(csc(f*x 
+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(5/2)-10*arctan(1/(1-cos(f*x 
+e))*(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)+1-cos(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(5 
/2)-5*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)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(5/2)-10*a 
rctan(1/(1-cos(f*x+e))*(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)-1+cos(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x 
+e)+cot(f*x+e))^(5/2)+112*csc(f*x+e)^5*(1-cos(f*x+e))^5-40*csc(f*x+e)^3*(1 
-cos(f*x+e))^3)/(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)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(5/2)*2^(1 
/2)
 

3.2.91.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.91 \[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=-\frac {5 \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 5 i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (i \, f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 5 i \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 5 \, f \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-f^{3} \left (-\frac {d^{2}}{f^{4}}\right )^{\frac {3}{4}} + d \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 4 \, {\left (\tan \left (f x + e\right )^{3} - 5 \, \tan \left (f x + e\right )\right )} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{10 \, f} \]

input

integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^4,x, algorithm="fricas")
 

output

-1/10*(5*f*(-d^2/f^4)^(1/4)*log(f^3*(-d^2/f^4)^(3/4) + d*sqrt(d/tan(f*x + 
e))) - 5*I*f*(-d^2/f^4)^(1/4)*log(I*f^3*(-d^2/f^4)^(3/4) + d*sqrt(d/tan(f* 
x + e))) + 5*I*f*(-d^2/f^4)^(1/4)*log(-I*f^3*(-d^2/f^4)^(3/4) + d*sqrt(d/t 
an(f*x + e))) - 5*f*(-d^2/f^4)^(1/4)*log(-f^3*(-d^2/f^4)^(3/4) + d*sqrt(d/ 
tan(f*x + e))) - 4*(tan(f*x + e)^3 - 5*tan(f*x + e))*sqrt(d/tan(f*x + e))) 
/f
 

3.2.91.6 Sympy [F]

\[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=\int \sqrt {d \cot {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )}\, dx \]

input

integrate((d*cot(f*x+e))**(1/2)*tan(f*x+e)**4,x)
 

output

Integral(sqrt(d*cot(e + f*x))*tan(e + f*x)**4, x)
 

3.2.91.7 Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.89 \[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=-\frac {d^{5} {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{d^{4}} - \frac {8 \, {\left (d^{2} - \frac {5 \, d^{2}}{\tan \left (f x + e\right )^{2}}\right )}}{d^{4} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, f} \]

input

integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^4,x, algorithm="maxima")
 

output

-1/20*d^5*(5*(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)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d) + sqrt(2)*log(-sqrt( 
2)*sqrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d))/d^4 - 8*(d^ 
2 - 5*d^2/tan(f*x + e)^2)/(d^4*(d/tan(f*x + e))^(5/2)))/f
 

3.2.91.8 Giac [F]

\[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=\int { \sqrt {d \cot \left (f x + e\right )} \tan \left (f x + e\right )^{4} \,d x } \]

input

integrate((d*cot(f*x+e))^(1/2)*tan(f*x+e)^4,x, algorithm="giac")
 

output

integrate(sqrt(d*cot(f*x + e))*tan(f*x + e)^4, x)
 

3.2.91.9 Mupad [B] (verification not implemented)

Time = 2.81 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.42 \[ \int \sqrt {d \cot (e+f x)} \tan ^4(e+f x) \, dx=\frac {\frac {2\,d^3}{5}-\frac {2\,d^3}{{\mathrm {tan}\left (e+f\,x\right )}^2}}{f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f}+\frac {{\left (-1\right )}^{1/4}\,\sqrt {d}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f} \]

input

int(tan(e + f*x)^4*(d*cot(e + f*x))^(1/2),x)
 

output

((2*d^3)/5 - (2*d^3)/tan(e + f*x)^2)/(f*(d/tan(e + f*x))^(5/2)) - ((-1)^(1 
/4)*d^(1/2)*atan(((-1)^(1/4)*(d/tan(e + f*x))^(1/2))/d^(1/2)))/f + ((-1)^( 
1/4)*d^(1/2)*atanh(((-1)^(1/4)*(d/tan(e + f*x))^(1/2))/d^(1/2)))/f