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

Optimal result

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

[Out]

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)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(204\) vs. \(2(92)=184\).

Time = 0.18 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.22, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {6851, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=-\frac {3 \sec ^2(x) \arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \sec ^2(x) \arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {3 \sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {3 \sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)} \]

[In]

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

[Out]

-1/2*(Cot[x]*Sqrt[Sin[x]^4*Tan[x]]) - (3*ArcTan[1 - Sqrt[2]*Sqrt[Tan[x]]]*Sec[x]^2*Sqrt[Sin[x]^4*Tan[x]])/(4*S
qrt[2]*Tan[x]^(5/2)) + (3*ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]]*Sec[x]^2*Sqrt[Sin[x]^4*Tan[x]])/(4*Sqrt[2]*Tan[x]^(
5/2)) + (3*Log[1 - Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]*Sec[x]^2*Sqrt[Sin[x]^4*Tan[x]])/(8*Sqrt[2]*Tan[x]^(5/2)) - (
3*Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]*Sec[x]^2*Sqrt[Sin[x]^4*Tan[x]])/(8*Sqrt[2]*Tan[x]^(5/2))

Rule 210

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 294

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 6851

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {\frac {x^5}{\left (1+x^2\right )^2}}}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {\left (\sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {x^{5/2}}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )}{\tan ^{\frac {5}{2}}(x)} \\ & = -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (x)\right )}{4 \tan ^{\frac {5}{2}}(x)} \\ & = -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{2 \tan ^{\frac {5}{2}}(x)} \\ & = -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}-\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{4 \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{4 \tan ^{\frac {5}{2}}(x)} \\ & = -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)} \\ & = -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {3 \log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {3 \log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)} \\ & = -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {3 \log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72 \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=-\frac {1}{8} \csc ^3(x) \left (3 \arcsin (\cos (x)-\sin (x))+3 \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+2 \sin (x) \sqrt {\sin (2 x)}\right ) \sqrt {\sin (2 x)} \sqrt {\sin ^4(x) \tan (x)} \]

[In]

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

[Out]

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(73)=146\).

Time = 7.83 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.25

method	result	size
default	\(\frac {\left (4 \cos \left (x \right ) \sin \left (x \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+4 \sqrt {2}\, \sqrt {-\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )-3 \ln \left (-\frac {\cos \left (x \right ) \cot \left (x \right )-2 \cot \left (x \right )-2 \sin \left (x \right ) \sqrt {-\left (\cot ^{3}\left (x \right )\right )+3 \csc \left (x \right ) \left (\cot ^{2}\left (x \right )\right )-3 \cot \left (x \right ) \left (\csc ^{2}\left (x \right )\right )+\csc ^{3}\left (x \right )-\csc \left (x \right )+\cot \left (x \right )}-2 \cos \left (x \right )-\sin \left (x \right )+\csc \left (x \right )+2}{-1+\cos \left (x \right )}\right )+3 \ln \left (-\frac {\cos \left (x \right ) \cot \left (x \right )-2 \cot \left (x \right )+2 \sin \left (x \right ) \sqrt {-\left (\cot ^{3}\left (x \right )\right )+3 \csc \left (x \right ) \left (\cot ^{2}\left (x \right )\right )-3 \cot \left (x \right ) \left (\csc ^{2}\left (x \right )\right )+\csc ^{3}\left (x \right )-\csc \left (x \right )+\cot \left (x \right )}-2 \cos \left (x \right )-\sin \left (x \right )+\csc \left (x \right )+2}{-1+\cos \left (x \right )}\right )+6 \arctan \left (\frac {-\sqrt {2}\, \sqrt {-\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )\right ) \sqrt {\left (\sin ^{4}\left (x \right )\right ) \tan \left (x \right )}\, \cos \left (x \right ) \sqrt {32}}{64 \left (-1+\cos \left (x \right )\right ) \left (\cos \left (x \right )+1\right )^{2} \sqrt {-\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(299\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 628, normalized size of antiderivative = 6.83 \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/64*(-(3*I - 3)*sqrt(2)*log((2*I*cos(x)^4 - 3*I*cos(x)^2 + 2*(cos(x)^3 - cos(x))*sin(x) + ((I + 1)*sqrt(2)*co
s(x)^2 - (I - 1)*sqrt(2)*cos(x)*sin(x))*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) + I)/(cos(x)^2 - 1))*s
in(x) + (3*I - 3)*sqrt(2)*log((2*I*cos(x)^4 - 3*I*cos(x)^2 + 2*(cos(x)^3 - cos(x))*sin(x) + (-(I + 1)*sqrt(2)*
cos(x)^2 + (I - 1)*sqrt(2)*cos(x)*sin(x))*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) + I)/(cos(x)^2 - 1))
*sin(x) + (3*I + 3)*sqrt(2)*log((-2*I*cos(x)^4 + 3*I*cos(x)^2 + 2*(cos(x)^3 - cos(x))*sin(x) + (-(I - 1)*sqrt(
2)*cos(x)^2 + (I + 1)*sqrt(2)*cos(x)*sin(x))*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) - I)/(cos(x)^2 -
1))*sin(x) - (3*I + 3)*sqrt(2)*log((-2*I*cos(x)^4 + 3*I*cos(x)^2 + 2*(cos(x)^3 - cos(x))*sin(x) + ((I - 1)*sqr
t(2)*cos(x)^2 - (I + 1)*sqrt(2)*cos(x)*sin(x))*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) - I)/(cos(x)^2
- 1))*sin(x) + (3*I + 3)*sqrt(2)*log((cos(x)^2 + ((I + 1)*sqrt(2)*cos(x)^2 - (I - 1)*sqrt(2)*cos(x)*sin(x))*sq
rt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)) - 1)/(cos(x)^2 - 1))*sin(x) - (3*I - 3)*sqrt(2)*log((cos(x)^2 +
(-(I - 1)*sqrt(2)*cos(x)^2 + (I + 1)*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) + (3*I - 3)*sqrt(2)*log((cos(x)^2 + ((I - 1)*sqrt(2)*cos(x)^2 - (I + 1)*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) - (3*I + 3)*sqrt(2)*log
((cos(x)^2 + (-(I + 1)*sqrt(2)*cos(x)^2 + (I - 1)*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) - 32*sqrt((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x))*cos(x))/sin(x)

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(sin(x)**5/cos(x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(sin(x)^5/cos(x)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(sin(x)^5/cos(x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((sin(x)^5/cos(x))^(1/2), x)