3.5.17 \(\int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx\) [417]

Optimal. Leaf size=364 \[ -2 \sqrt {2} \coth ^{-1}\left (\frac {\cos (x) (\cos (x)+\sin (x))}{\sqrt {2} \sqrt {\cos ^3(x) \sin (x)}}\right )+\sqrt [4]{2} \coth ^{-1}\left (\frac {\cos (x) \left (\sqrt {2} \cos (x)+\sin (x)\right )}{2^{3/4} \sqrt {\cos ^3(x) \sin (x)}}\right )-\sqrt [4]{2} \coth ^{-1}\left (\frac {\sqrt {2}+\tan (x)}{2^{3/4} \sqrt {\tan (x)}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\cos (x) (\cos (x)-\sin (x))}{\sqrt {2} \sqrt {\cos ^3(x) \sin (x)}}\right )+\sqrt [4]{2} \tan ^{-1}\left (\frac {\cos (x) \left (\sqrt {2} \cos (x)-\sin (x)\right )}{2^{3/4} \sqrt {\cos ^3(x) \sin (x)}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt {2}-\tan (x)}{2^{3/4} \sqrt {\tan (x)}}\right )+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {1}{4} \csc ^2(x) \log \left (1+\cos ^2(x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{2} \csc ^2(x) \log (\sin (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {4}{\sqrt {\tan (x)}}-\frac {1}{4} \csc ^2(x) \log \left (1+\cos ^2(x)\right ) \sqrt {\cos (x) \sin ^3(x)} \sqrt {\tan (x)}+\frac {1}{2} \csc ^2(x) \log (\sin (x)) \sqrt {\cos (x) \sin ^3(x)} \sqrt {\tan (x)} \]

[Out]

2^(1/4)*arccoth(1/2*cos(x)*(sin(x)+cos(x)*2^(1/2))*2^(1/4)/(cos(x)^3*sin(x))^(1/2))-2^(1/4)*arccoth(1/2*(2^(1/
2)+tan(x))*2^(1/4)/tan(x)^(1/2))+2^(1/4)*arctan(1/2*cos(x)*(-sin(x)+cos(x)*2^(1/2))*2^(1/4)/(cos(x)^3*sin(x))^
(1/2))-2^(1/4)*arctan(1/2*(2^(1/2)-tan(x))*2^(1/4)/tan(x)^(1/2))-2*arccoth(1/2*cos(x)*(cos(x)+sin(x))*2^(1/2)/
(cos(x)^3*sin(x))^(1/2))*2^(1/2)-2*arctan(1/2*cos(x)*(cos(x)-sin(x))*2^(1/2)/(cos(x)^3*sin(x))^(1/2))*2^(1/2)+
4*csc(x)*sec(x)*(cos(x)^3*sin(x))^(1/2)+1/4*csc(x)^2*ln(1+cos(x)^2)*sec(x)^2*(cos(x)^3*sin(x))^(1/2)*(cos(x)*s
in(x)^3)^(1/2)+1/2*csc(x)^2*ln(sin(x))*sec(x)^2*(cos(x)^3*sin(x))^(1/2)*(cos(x)*sin(x)^3)^(1/2)+4/tan(x)^(1/2)
-1/4*csc(x)^2*ln(1+cos(x)^2)*(cos(x)*sin(x)^3)^(1/2)*tan(x)^(1/2)+1/2*csc(x)^2*ln(sin(x))*(cos(x)*sin(x)^3)^(1
/2)*tan(x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 3.27, antiderivative size = 665, normalized size of antiderivative = 1.83, number of steps used = 66, number of rules used = 21, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {6857, 6874, 6851, 331, 335, 303, 1176, 631, 210, 1179, 642, 477, 493, 6865, 15, 29, 272, 36, 31, 266, 455} \begin {gather*} -\sqrt [4]{2} \text {ArcTan}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \text {ArcTan}\left (\sqrt [4]{2} \sqrt {\tan (x)}+1\right )+\frac {\sqrt [4]{2} \sec ^2(x) \text {ArcTan}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sqrt [4]{2} \sec ^2(x) \text {ArcTan}\left (\sqrt [4]{2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {2 \sqrt {2} \sec ^2(x) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {2 \sqrt {2} \sec ^2(x) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {4}{\sqrt {\tan (x)}}+\frac {\log \left (\tan (x)-2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right )}{2^{3/4}}-\frac {\log \left (\tan (x)+2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\sin (x) \cos ^3(x)}+\frac {\sqrt {2} \sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sqrt {2} \sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-\frac {\sec ^2(x) \log \left (\tan (x)-2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right ) \sqrt {\sin (x) \cos ^3(x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {\sec ^2(x) \log \left (\tan (x)+2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right ) \sqrt {\sin (x) \cos ^3(x)}}{2^{3/4} \sqrt {\tan (x)}}-\frac {1}{2} \csc ^2(x) \sec ^2(x) \log \left (\sec ^2(x)\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)}+\frac {\sec ^2(x) \log (\tan (x)) \sqrt {\sin ^3(x) \cos (x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\sec ^2(x) \log \left (\tan ^2(x)+2\right ) \sqrt {\sin ^3(x) \cos (x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {1}{4} \csc ^2(x) \sec ^2(x) \log \left (\tan ^2(x)+2\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)}+\csc ^2(x) \sec ^2(x) \log \left (\sqrt {\tan (x)}\right ) \sqrt {\sin (x) \cos ^3(x)} \sqrt {\sin ^3(x) \cos (x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(Sqrt[Cos[x]*Sin[x]^3] - 2*Sin[2*x])/(-Sqrt[Cos[x]^3*Sin[x]] + Sqrt[Tan[x]]),x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 331

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

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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 493

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

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]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx &=\text {Subst}\left (\int \frac {\sqrt {\frac {x^3}{\left (1+x^2\right )^2}}-\frac {4 x}{1+x^2}}{\left (1+x^2\right ) \left (\sqrt {x}-\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {4 x}{\left (1+x^2\right )^2 \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )}-\frac {\sqrt {\frac {x^3}{\left (1+x^2\right )^2}}}{\left (1+x^2\right ) \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )}\right ) \, dx,x,\tan (x)\right )\\ &=4 \text {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2 \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )-\text {Subst}\left (\int \frac {\sqrt {\frac {x^3}{\left (1+x^2\right )^2}}}{\left (1+x^2\right ) \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )\\ &=4 \text {Subst}\left (\int \left (-\frac {1}{2 x^{3/2}}-\frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{2 x^2}+\frac {\sqrt {x}}{2 \left (2+x^2\right )}+\frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{2 \left (2+x^2\right )}\right ) \, dx,x,\tan (x)\right )-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{\left (1+x^2\right )^2 \left (-\sqrt {x}+\sqrt {\frac {x}{\left (1+x^2\right )^2}}\right )} \, dx,x,\tan (x)\right )}{\tan ^{\frac {3}{2}}(x)}\\ &=\frac {4}{\sqrt {\tan (x)}}-2 \text {Subst}\left (\int \frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{x^2} \, dx,x,\tan (x)\right )+2 \text {Subst}\left (\int \frac {\sqrt {x}}{2+x^2} \, dx,x,\tan (x)\right )+2 \text {Subst}\left (\int \frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}}}{2+x^2} \, dx,x,\tan (x)\right )-\frac {\left (2 \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^4\right )^2 \left (-\sqrt {x^2}+\sqrt {\frac {x^2}{\left (1+x^4\right )^2}}\right )} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}\\ &=\frac {4}{\sqrt {\tan (x)}}+4 \text {Subst}\left (\int \frac {x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )-\frac {\left (2 \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 \sqrt {x^2}}-\frac {\sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{2 x^2}+\frac {\left (x^2\right )^{3/2}}{2 \left (2+x^4\right )}+\frac {x^2 \sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{2 \left (2+x^4\right )}\right ) \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}\\ &=4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {4}{\sqrt {\tan (x)}}-2 \text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )+2 \text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )+\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x^2}} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}+\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {\left (x^2\right )^{3/2}}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {\frac {x^2}{\left (1+x^4\right )^2}}}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}+\frac {\left (4 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right ) \left (2+x^4\right )} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\\ &=4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {4}{\sqrt {\tan (x)}}+\frac {\text {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4}}+\frac {\text {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4}}+\left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{x \left (1+x^4\right )} \, dx,x,\sqrt {\tan (x)}\right )-\left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^4\right ) \left (2+x^4\right )} \, dx,x,\sqrt {\tan (x)}\right )+\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}-\frac {\left (\sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {x^3}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\tan ^{\frac {3}{2}}(x)}+2 \frac {\left (4 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (4 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\text {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )+\text {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )\\ &=\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}+\sqrt [4]{2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{2} \sqrt {\tan (x)}\right )-\sqrt [4]{2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,\tan ^2(x)\right )-\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{(1+x) (2+x)} \, dx,x,\tan ^2(x)\right )+2 \left (-\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\right )+\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\\ &=-\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}+\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan ^2(x)\right )-2 \left (\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^2(x)\right )\right )+\frac {1}{4} \left (\csc ^2(x) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,\tan ^2(x)\right )-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4} \sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2^{3/4} \sqrt {\tan (x)}}+2 \left (\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {2} \sqrt {\tan (x)}}\right )\\ &=-\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\csc ^2(x) \log (\cos (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{2} \csc ^2(x) \log (\tan (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{4} \csc ^2(x) \log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}-\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}-\frac {\left (\sqrt [4]{2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {\left (\sqrt [4]{2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+2 \left (\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\sqrt {2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\left (\sqrt {2} \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}\right )\\ &=-\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\csc ^2(x) \log (\cos (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{2} \csc ^2(x) \log (\tan (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{4} \csc ^2(x) \log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+2 \left (-\frac {\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}\right )+\frac {\log (\tan (x)) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\log \left (2+\tan ^2(x)\right ) \sec ^2(x) \sqrt {\cos (x) \sin ^3(x)}}{4 \tan ^{\frac {3}{2}}(x)}+\frac {4}{\sqrt {\tan (x)}}+\frac {\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}-\frac {\sqrt [4]{2} \tan ^{-1}\left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}-\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 7.66, size = 385, normalized size = 1.06 \begin {gather*} 4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}-\frac {\cos (x) \csc \left (\frac {x}{2}\right ) \left (4 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )-2 \log \left (\tan \left (\frac {x}{2}\right )\right )-\log \left (1+\tan ^4\left (\frac {x}{2}\right )\right )\right ) \sec \left (\frac {x}{2}\right ) \sqrt {\cos (x) \sin ^3(x)}}{8 \sqrt {\cos ^3(x) \sin (x)}}-\frac {(1+i) \left ((4+4 i) \Pi \left (-i;\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )-(4+4 i) \Pi \left (i;\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )+\sqrt [4]{-1} \left (-\Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )+\Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )+\Pi \left ((-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {\tan \left (\frac {x}{2}\right )}\right )\right |-1\right )\right )\right ) \sec ^4\left (\frac {x}{2}\right ) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\cos (x) \sec ^2\left (\frac {x}{2}\right )} \sqrt {\tan \left (\frac {x}{2}\right )} \left (-1+\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {4}{\sqrt {\tan (x)}}+\frac {1}{4} \csc ^2(x) \left (2 \log (\tan (x))-\log \left (2+\tan ^2(x)\right )\right ) \sqrt {\cos (x) \sin ^3(x)} \sqrt {\tan (x)}+\frac {4 \sqrt {2} \cos ^2(x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {2 \sin ^2(x)}{3+\cos (2 x)}\right ) \tan ^{\frac {3}{2}}(x)}{3 (3+\cos (2 x))^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[Cos[x]*Sin[x]^3] - 2*Sin[2*x])/(-Sqrt[Cos[x]^3*Sin[x]] + Sqrt[Tan[x]]),x]

[Out]

4*Csc[x]*Sec[x]*Sqrt[Cos[x]^3*Sin[x]] - (Cos[x]*Csc[x/2]*(4*Log[Sec[x/2]^2] - 2*Log[Tan[x/2]] - Log[1 + Tan[x/
2]^4])*Sec[x/2]*Sqrt[Cos[x]*Sin[x]^3])/(8*Sqrt[Cos[x]^3*Sin[x]]) - ((1 + I)*((4 + 4*I)*EllipticPi[-I, ArcSin[S
qrt[Tan[x/2]]], -1] - (4 + 4*I)*EllipticPi[I, ArcSin[Sqrt[Tan[x/2]]], -1] + (-1)^(1/4)*(-EllipticPi[-(-1)^(1/4
), ArcSin[Sqrt[Tan[x/2]]], -1] + EllipticPi[(-1)^(1/4), ArcSin[Sqrt[Tan[x/2]]], -1] - EllipticPi[-(-1)^(3/4),
ArcSin[Sqrt[Tan[x/2]]], -1] + EllipticPi[(-1)^(3/4), ArcSin[Sqrt[Tan[x/2]]], -1]))*Sec[x/2]^4*Sqrt[Cos[x]^3*Si
n[x]])/(Sqrt[Cos[x]*Sec[x/2]^2]*Sqrt[Tan[x/2]]*(-1 + Tan[x/2]^2)) + 4/Sqrt[Tan[x]] + (Csc[x]^2*(2*Log[Tan[x]]
- Log[2 + Tan[x]^2])*Sqrt[Cos[x]*Sin[x]^3]*Sqrt[Tan[x]])/4 + (4*Sqrt[2]*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4
, 3/4, 7/4, (2*Sin[x]^2)/(3 + Cos[2*x])]*Tan[x]^(3/2))/(3*(3 + Cos[2*x])^(3/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 9.42, size = 22968, normalized size = 63.10

method	result	size
default	\(\text {Expression too large to display}\)	\(22968\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*sin(2*x)+(cos(x)*sin(x)^3)^(1/2))/(-(cos(x)^3*sin(x))^(1/2)+tan(x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*sin(2*x)+(cos(x)*sin(x)^3)^(1/2))/(-(cos(x)^3*sin(x))^(1/2)+tan(x)^(1/2)),x, algorithm="maxima")

[Out]

-2*integrate(-1/4*(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^(1/4)*(((((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x)
+ sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*cos(4*x) - (sqrt(2)*cos(3*x) + 2*sq
rt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*c
os(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(1/
2*arctan2(sin(x), -cos(x) + 1)) - ((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x)
- 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(4*x) + (sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) + sq
rt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(
2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*cos
(1/2*arctan2(sin(x), cos(x) + 1)) + (((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*
x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(4*x) + (sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) +
sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sq
rt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*cos(1/2*arctan2(sin(x), -cos(x) + 1)) +
((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*si
n(x))*cos(4*x) - (sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*sqrt(2)*sin(2*
x) - sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*
sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1))
)*cos(1/2*arctan2(sin(2*x), cos(2*x) + 1)) - ((((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt
(2)*sin(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(4*x) + (sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2
)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos
(2*x) - sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*cos(1/2*arctan2(sin(x), -cos(
x) + 1)) + ((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) +
sqrt(2)*sin(x))*cos(4*x) - (sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*sqrt
(2)*sin(2*x) - sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(2)*cos(x) - sqrt(2)*sin
(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*cos(1/2*arctan2(sin(x), co
s(x) + 1)) - (((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x)
+ sqrt(2)*sin(x))*cos(4*x) - (sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*s
qrt(2)*sin(2*x) - sqrt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(2)*cos(x) - sqrt(2)*
sin(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(1/2*arctan2(sin(x), -cos(x) + 1)) - ((sqrt(2)*cos(3*x) + 2
*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(4*x) + (sqrt(
2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*si
n(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(2)*cos(x) + sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) + sqrt
(2)*sin(x))*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1)))*sin(1/2*arctan2(sin(2
*x), cos(2*x) + 1)))/((((2*(2*cos(2*x) + cos(x))*cos(3*x) + cos(3*x)^2 + 4*cos(2*x)^2 + 4*cos(2*x)*cos(x) + co
s(x)^2 + 2*(2*sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + 4*sin(2*x)^2 + 4*sin(2*x)*sin(x) + sin(x)^2)*cos(1/2*
arctan2(sin(x), -cos(x) + 1))^2 + (2*(2*cos(2*x) + cos(x))*cos(3*x) + cos(3*x)^2 + 4*cos(2*x)^2 + 4*cos(2*x)*c
os(x) + cos(x)^2 + 2*(2*sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + 4*sin(2*x)^2 + 4*sin(2*x)*sin(x) + sin(x)^2
)*sin(1/2*arctan2(sin(x), -cos(x) + 1))^2)*cos(1/2*arctan2(sin(x), cos(x) + 1))^2 + ((2*(2*cos(2*x) + cos(x))*
cos(3*x) + cos(3*x)^2 + 4*cos(2*x)^2 + 4*cos(2*x)*cos(x) + cos(x)^2 + 2*(2*sin(2*x) + sin(x))*sin(3*x) + sin(3
*x)^2 + 4*sin(2*x)^2 + 4*sin(2*x)*sin(x) + sin(x)^2)*cos(1/2*arctan2(sin(x), -cos(x) + 1))^2 + (2*(2*cos(2*x)
+ cos(x))*cos(3*x) + cos(3*x)^2 + 4*cos(2*x)^2 + 4*cos(2*x)*cos(x) + cos(x)^2 + 2*(2*sin(2*x) + sin(x))*sin(3*
x) + sin(3*x)^2 + 4*sin(2*x)^2 + 4*sin(2*x)*sin(x) + sin(x)^2)*sin(1/2*arctan2(sin(x), -cos(x) + 1))^2)*sin(1/
2*arctan2(sin(x), cos(x) + 1))^2)*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*cos(x) +
1)^(1/4)), x) + 2*integrate(1/4*(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^(1/4)*(((((sqrt(2)*cos(3*x) - 2*sq
rt(2)*cos(2*x) + sqrt(2)*cos(x) + sqrt(2)*sin(3*x) - 2*sqrt(2)*sin(2*x) + sqrt(2)*sin(x))*cos(4*x) - (sqrt(2)*
cos(3*x) - 2*sqrt(2)*cos(2*x) + sqrt(2)*cos(x) - sqrt(2)*sin(3*x) + 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*sin(4
*x) - sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) - s...

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*sin(2*x)+(cos(x)*sin(x)^3)^(1/2))/(-(cos(x)^3*sin(x))^(1/2)+tan(x)^(1/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   not invertible

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*sin(2*x)+(cos(x)*sin(x)**3)**(1/2))/(-(cos(x)**3*sin(x))**(1/2)+tan(x)**(1/2)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*sin(2*x)+(cos(x)*sin(x)^3)^(1/2))/(-(cos(x)^3*sin(x))^(1/2)+tan(x)^(1/2)),x, algorithm="giac")

[Out]

integrate(-(sqrt(cos(x)*sin(x)^3) - 2*sin(2*x))/(sqrt(cos(x)^3*sin(x)) - sqrt(tan(x))), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {2\,\sin \left (2\,x\right )-\sqrt {\cos \left (x\right )\,{\sin \left (x\right )}^3}}{\sqrt {\mathrm {tan}\left (x\right )}-\sqrt {{\cos \left (x\right )}^3\,\sin \left (x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*sin(2*x) - (cos(x)*sin(x)^3)^(1/2))/(tan(x)^(1/2) - (cos(x)^3*sin(x))^(1/2)),x)

[Out]

-int((2*sin(2*x) - (cos(x)*sin(x)^3)^(1/2))/(tan(x)^(1/2) - (cos(x)^3*sin(x))^(1/2)), x)

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