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

Optimal. Leaf size=92 \[ -\frac{1}{2} \cot (x) \sqrt{\sin ^4(x) \tan (x)}+\frac{3 \tan ^{-1}\left (\frac{(1-\cot (x)) \csc ^2(x) \sqrt{\sin ^4(x) \tan (x)}}{\sqrt{2}}\right )}{4 \sqrt{2}}+\frac{3 \log \left (\sin (x)+\cos (x)-\sqrt{2} \cot (x) \csc (x) \sqrt{\sin ^4(x) \tan (x)}\right )}{4 \sqrt{2}} \]

[Out]

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

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Rubi [B]  time = 0.239033, antiderivative size = 204, normalized size of antiderivative = 2.22, number of steps used = 13, number of rules used = 9, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818, Rules used = {6719, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{1}{2} \cot (x) \sqrt{\sin ^4(x) \tan (x)}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\sin ^4(x) \tan (x)}}{4 \sqrt{2} \tan ^{\frac{5}{2}}(x)}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (x)}+1\right ) \sec ^2(x) \sqrt{\sin ^4(x) \tan (x)}}{4 \sqrt{2} \tan ^{\frac{5}{2}}(x)}+\frac{3 \sec ^2(x) \log \left (\tan (x)-\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin ^4(x) \tan (x)}}{8 \sqrt{2} \tan ^{\frac{5}{2}}(x)}-\frac{3 \sec ^2(x) \log \left (\tan (x)+\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin ^4(x) \tan (x)}}{8 \sqrt{2} \tan ^{\frac{5}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cot[x]*Sqrt[Sin[x]^4*Tan[x]])/2 - (3*ArcTan[1 - Sqrt[2]*Sqrt[Tan[x]]]*Sec[x]^2*Sqrt[Sin[x]^4*Tan[x]])/(4*Sqr
t[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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[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]

Rule 288

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 329

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 297

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 1162

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 617

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

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 628

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]

Rubi steps

\begin{align*} \int \sqrt{\sin ^4(x) \tan (x)} \, dx &=\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\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 \tan ^{-1}\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]  time = 0.0742365, size = 66, normalized size = 0.72 \[ -\frac{1}{8} \sqrt{\sin (2 x)} \csc ^3(x) \sqrt{\sin ^4(x) \tan (x)} \left (2 \sin (x) \sqrt{\sin (2 x)}+3 \sin ^{-1}(\cos (x)-\sin (x))+3 \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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]])*Sqr
t[Sin[2*x]]*Sqrt[Sin[x]^4*Tan[x]])/8

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Maple [C]  time = 0.141, size = 310, normalized size = 3.4 \begin{align*} -{\frac{\sqrt{32} \left ( \cos \left ( x \right ) -1 \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{32\, \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( -3\,i\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}\sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +3\,i\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}\sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -3\,\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}\sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},1/2+i/2,1/2\,\sqrt{2} \right ) -3\,\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}\sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }}},1/2-i/2,1/2\,\sqrt{2} \right ) +2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{2}-2\,\cos \left ( x \right ) \sqrt{2} \right ) \sqrt{{\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{\cos \left ( x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*32^(1/2)*(cos(x)-1)*(-3*I*((cos(x)-1)/sin(x))^(1/2)*((sin(x)-1+cos(x))/sin(x))^(1/2)*((1-cos(x)+sin(x))/
sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*((cos(x)-1)/sin(x))^(1/2)
*((sin(x)-1+cos(x))/sin(x))^(1/2)*((1-cos(x)+sin(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2)
,1/2-1/2*I,1/2*2^(1/2))-3*((cos(x)-1)/sin(x))^(1/2)*((sin(x)-1+cos(x))/sin(x))^(1/2)*((1-cos(x)+sin(x))/sin(x)
)^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-3*((cos(x)-1)/sin(x))^(1/2)*((sin(x
)-1+cos(x))/sin(x))^(1/2)*((1-cos(x)+sin(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2
*I,1/2*2^(1/2))+2*cos(x)^2*2^(1/2)-2*cos(x)*2^(1/2))*(cos(x)+1)^2*(sin(x)^5/cos(x))^(1/2)/sin(x)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{\sin \left (x\right )^{5}}{\cos \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(x)^5/cos(x)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{\sin ^{5}{\left (x \right )}}{\cos{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sin(x)**5/cos(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{\sin \left (x\right )^{5}}{\cos \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(x)^5/cos(x)), x)