\(\int \sqrt {\sin ^4(x) \tan (x)} \, dx\) [413]

Optimal result

Integrand size = 11, antiderivative size = 92 \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\frac {3 \arctan \left (\frac {(1-\cot (x)) \csc ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {3 \log \left (\cos (x)+\sin (x)-\sqrt {2} \cot (x) \csc (x) \sqrt {\sin ^4(x) \tan (x)}\right )}{4 \sqrt {2}}-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)} \] Output:

3/8*arctan(1/2*(1-cot(x))*csc(x)^2*(sin(x)^4*tan(x))^(1/2)*2^(1/2))*2^(1/2 
)+3/8*ln(cos(x)+sin(x)-cot(x)*csc(x)*2^(1/2)*(sin(x)^4*tan(x))^(1/2))*2^(1 
/2)-1/2*cot(x)*(sin(x)^4*tan(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72 \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=-\frac {1}{8} \csc ^3(x) \left (3 \arcsin (\cos (x)-\sin (x))+3 \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+2 \sin (x) \sqrt {\sin (2 x)}\right ) \sqrt {\sin (2 x)} \sqrt {\sin ^4(x) \tan (x)} \] Input:

Integrate[Sqrt[Sin[x]^4*Tan[x]],x]
 

Output:

-1/8*(Csc[x]^3*(3*ArcSin[Cos[x] - Sin[x]] + 3*Log[Cos[x] + Sin[x] + Sqrt[S 
in[2*x]]] + 2*Sin[x]*Sqrt[Sin[2*x]])*Sqrt[Sin[2*x]]*Sqrt[Sin[x]^4*Tan[x]])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.75, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.182, Rules used = {3042, 4889, 7270, 252, 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.

\(\displaystyle \int \sqrt {\sin ^4(x) \tan (x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {\sin (x)^4 \tan (x)}dx\)

\(\Big \downarrow \) 4889

\(\displaystyle \int \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}}}{\tan ^2(x)+1}d\tan (x)\)

\(\Big \downarrow \) 7270

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \int \frac {\tan ^{\frac {5}{2}}(x)}{\left (\tan ^2(x)+1\right )^2}d\tan (x)}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 266

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \int \frac {\tan (x)}{\tan ^2(x)+1}d\sqrt {\tan (x)}-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {\tan (x)+1}{\tan ^2(x)+1}d\sqrt {\tan (x)}-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\sqrt {\tan (x)}\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}+\frac {1}{2} \int \frac {1}{\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\sqrt {\tan (x)}\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (x)-1}d\left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (x)-1}d\left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\sqrt {\tan (x)}\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\sqrt {\tan (x)}\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (x)}}{\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}\right )\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (x)}}{\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}\right )\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (x)}}{\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (x)}+1}{\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1}d\sqrt {\tan (x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}\right )\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt {\frac {\tan ^5(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}\right )\right )-\frac {\tan ^{\frac {3}{2}}(x)}{2 \left (\tan ^2(x)+1\right )}\right )}{\tan ^{\frac {5}{2}}(x)}\)

Input:

Int[Sqrt[Sin[x]^4*Tan[x]],x]
 

Output:

(Sqrt[Tan[x]^5/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2)*((3*((-(ArcTan[1 - Sqrt[2] 
*Sqrt[Tan[x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]]/Sqrt[2])/2 + (L 
og[1 - Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[T 
an[x]] + Tan[x]]/(2*Sqrt[2]))/2))/2 - Tan[x]^(3/2)/(2*(1 + Tan[x]^2))))/Ta 
n[x]^(5/2)
 

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

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 3042

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

rule 4889

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors 
[Tan[v], x]}, Simp[d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] &&  !MatchQ[ 
u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I 
ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
 

rule 7270

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p 
]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])))   Int[u*v 
^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !Free 
Q[v, x] &&  !FreeQ[w, x]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(73)=146\).

Time = 5.82 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.54

method	result	size
default	\(\frac {\cot \left (x \right ) \csc \left (x \right ) \left (3 \ln \left (-\frac {\cot \left (x \right ) \cos \left (x \right )-2 \cot \left (x \right )-2 \sin \left (x \right ) \sqrt {-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}-2 \cos \left (x \right )-\sin \left (x \right )+\csc \left (x \right )+2}{-1+\cos \left (x \right )}\right )-3 \ln \left (-\frac {\cot \left (x \right ) \cos \left (x \right )-2 \cot \left (x \right )+2 \sin \left (x \right ) \sqrt {-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}-2 \cos \left (x \right )-\sin \left (x \right )+\csc \left (x \right )+2}{-1+\cos \left (x \right )}\right )-6 \arctan \left (\frac {-\sin \left (x \right ) \sqrt {-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )+6 \arctan \left (\frac {\sin \left (x \right ) \sqrt {-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )+\left (-4 \cos \left (x \right )-4\right ) \sin \left (x \right ) \sqrt {-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}\right ) \sqrt {\tan \left (x \right ) \left (-1+\cos \left (x \right )\right )^{2} \left (1+\cos \left (x \right )\right )^{2}}\, \sqrt {32}}{64 \sqrt {-\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}\, \left (1+\cos \left (x \right )\right )}\)	\(234\)

Input:

int((sin(x)^5/cos(x))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/64*cot(x)*csc(x)*(3*ln(-(cot(x)*cos(x)-2*cot(x)-2*sin(x)*(-2*cos(x)*sin( 
x)/(1+cos(x))^2)^(1/2)-2*cos(x)-sin(x)+csc(x)+2)/(-1+cos(x)))-3*ln(-(cot(x 
)*cos(x)-2*cot(x)+2*sin(x)*(-2*cos(x)*sin(x)/(1+cos(x))^2)^(1/2)-2*cos(x)- 
sin(x)+csc(x)+2)/(-1+cos(x)))-6*arctan((-sin(x)*(-2*cos(x)*sin(x)/(1+cos(x 
))^2)^(1/2)+cos(x)-1)/(-1+cos(x)))+6*arctan((sin(x)*(-2*cos(x)*sin(x)/(1+c 
os(x))^2)^(1/2)+cos(x)-1)/(-1+cos(x)))+(-4*cos(x)-4)*sin(x)*(-2*cos(x)*sin 
(x)/(1+cos(x))^2)^(1/2))*(tan(x)*(-1+cos(x))^2*(1+cos(x))^2)^(1/2)/(-cos(x 
)*sin(x)/(1+cos(x))^2)^(1/2)/(1+cos(x))*32^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (71) = 142\).

Time = 0.11 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.54 \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx =\text {Too large to display} \] Input:

integrate((sin(x)^5/cos(x))^(1/2),x, algorithm="fricas")
 

Output:

1/32*(6*sqrt(2)*arctan(-sqrt(2)*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/co 
s(x))*cos(x)/(cos(x)^3 - (cos(x)^2 - 1)*sin(x) - cos(x)))*sin(x) + 3*sqrt( 
2)*arctan(1/2*(2*cos(x)^4 - 4*cos(x)^2 - 2*(cos(x)^3 - cos(x))*sin(x) + sq 
rt(2)*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) + 2)/(cos(x)^4 - 2*c 
os(x)^2 + (cos(x)^3 - cos(x))*sin(x) + 1))*sin(x) + 3*sqrt(2)*arctan(-1/2* 
(2*cos(x)^4 - 4*cos(x)^2 - 2*(cos(x)^3 - cos(x))*sin(x) - sqrt(2)*sqrt((co 
s(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) + 2)/(cos(x)^4 - 2*cos(x)^2 + (cos 
(x)^3 - cos(x))*sin(x) + 1))*sin(x) + 3*sqrt(2)*log((cos(x)^2 + 4*(cos(x)^ 
3 - cos(x))*sin(x) + 2*(sqrt(2)*cos(x)^2 + sqrt(2)*cos(x)*sin(x))*sqrt((co 
s(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) - 1)/(cos(x)^2 - 1))*sin(x) - 3*sq 
rt(2)*log((cos(x)^2 + 4*(cos(x)^3 - cos(x))*sin(x) - 2*(sqrt(2)*cos(x)^2 + 
 sqrt(2)*cos(x)*sin(x))*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) - 
1)/(cos(x)^2 - 1))*sin(x) - 16*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos 
(x))*cos(x))/sin(x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\int \sqrt {\frac {\sin ^{5}{\left (x \right )}}{\cos {\left (x \right )}}}\, dx \] Input:

integrate((sin(x)**5/cos(x))**(1/2),x)
 

Output:

Integral(sqrt(sin(x)**5/cos(x)), x)
 

Maxima [F]

\[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\int { \sqrt {\frac {\sin \left (x\right )^{5}}{\cos \left (x\right )}} \,d x } \] Input:

integrate((sin(x)^5/cos(x))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(sin(x)^5/cos(x)), x)
 

Giac [F]

\[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\int { \sqrt {\frac {\sin \left (x\right )^{5}}{\cos \left (x\right )}} \,d x } \] Input:

integrate((sin(x)^5/cos(x))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(sin(x)^5/cos(x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\int \sqrt {\frac {{\sin \left (x\right )}^5}{\cos \left (x\right )}} \,d x \] Input:

int((sin(x)^5/cos(x))^(1/2),x)
 

Output:

int((sin(x)^5/cos(x))^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {\cos \left (x \right )}\, \sin \left (x \right )^{2}}{\cos \left (x \right )}d x \] Input:

int((sin(x)^5/cos(x))^(1/2),x)
 

Output:

int((sqrt(sin(x))*sqrt(cos(x))*sin(x)**2)/cos(x),x)