3.417 \(\int \frac{\sqrt{\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt{\cos ^3(x) \sin (x)}+\sqrt{\tan (x)}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 4.73026, 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 = {6725, 6742, 6719, 325, 329, 297, 1162, 617, 204, 1165, 628, 466, 482, 6733, 15, 29, 266, 36, 31, 260, 444} \[ -\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt{\tan (x)}\right )+\sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt{\tan (x)}+1\right )+\frac{4}{\sqrt{\tan (x)}}+\frac{\log \left (\tan (x)-2^{3/4} \sqrt{\tan (x)}+\sqrt{2}\right )}{2^{3/4}}-\frac{\log \left (\tan (x)+2^{3/4} \sqrt{\tan (x)}+\sqrt{2}\right )}{2^{3/4}}+4 \csc (x) \sec (x) \sqrt{\sin (x) \cos ^3(x)}+\frac{\sqrt [4]{2} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{\sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt{\tan (x)}+1\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (x)}+1\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}+\frac{\sqrt{2} \sec ^2(x) \log \left (\tan (x)-\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{\sqrt{2} \sec ^2(x) \log \left (\tan (x)+\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{\sec ^2(x) \log \left (\tan (x)-2^{3/4} \sqrt{\tan (x)}+\sqrt{2}\right ) \sqrt{\sin (x) \cos ^3(x)}}{2^{3/4} \sqrt{\tan (x)}}+\frac{\sec ^2(x) \log \left (\tan (x)+2^{3/4} \sqrt{\tan (x)}+\sqrt{2}\right ) \sqrt{\sin (x) \cos ^3(x)}}{2^{3/4} \sqrt{\tan (x)}}-\frac{1}{2} \csc ^2(x) \sec ^2(x) \log \left (\sec ^2(x)\right ) \sqrt{\sin (x) \cos ^3(x)} \sqrt{\sin ^3(x) \cos (x)}+\frac{\sec ^2(x) \log (\tan (x)) \sqrt{\sin ^3(x) \cos (x)}}{2 \tan ^{\frac{3}{2}}(x)}-\frac{\sec ^2(x) \log \left (\tan ^2(x)+2\right ) \sqrt{\sin ^3(x) \cos (x)}}{4 \tan ^{\frac{3}{2}}(x)}+\frac{1}{4} \csc ^2(x) \sec ^2(x) \log \left (\tan ^2(x)+2\right ) \sqrt{\sin (x) \cos ^3(x)} \sqrt{\sin ^3(x) \cos (x)}+\csc ^2(x) \sec ^2(x) \log \left (\sqrt{\tan (x)}\right ) \sqrt{\sin (x) \cos ^3(x)} \sqrt{\sin ^3(x) \cos (x)} \]

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 6725

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 6742

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

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 325

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 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]

Rule 466

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 482

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 6733

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 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 266

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 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 31

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

Rule 260

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 444

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]

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 &=\operatorname{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 )\\ &=\operatorname{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 \operatorname{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 )-\operatorname{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 \operatorname{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 ) \operatorname{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 \operatorname{Subst}\left (\int \frac{\sqrt{\frac{x}{\left (1+x^2\right )^2}}}{x^2} \, dx,x,\tan (x)\right )+2 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{2+x^2} \, dx,x,\tan (x)\right )+2 \operatorname{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 ) \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{2+x^4} \, dx,x,\sqrt{\tan (x)}\right )+2 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{x^2}{2+x^4} \, dx,x,\sqrt{\tan (x)}\right )}{\sqrt{\tan (x)}}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-2^{3/4} x+x^2} \, dx,x,\sqrt{\tan (x)}\right )+\operatorname{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} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{2} \sqrt{\tan (x)}\right )-\sqrt [4]{2} \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 14.5279, size = 475, normalized size = 1.3 \[ -\frac{2 \left (2-\sin ^2(x)\right ) \cos ^2(x)^{3/4} \tan ^{\frac{3}{2}}(x) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{\sin ^2(x)}{2 \left (1-\frac{\sin ^2(x)}{2}\right )}\right )}{3 \left (1-\frac{\sin ^2(x)}{2}\right )^{3/4} \left (\sin ^2(x)-2\right )}+\frac{4}{\sqrt{\tan (x)}}+\sqrt{2 \sin (2 x)+\sin (4 x)} \left (\sqrt{2} \tan (x)+\sqrt{2} \cot (x)\right )+\frac{(1+i) \sec ^4\left (\frac{x}{2}\right ) \sqrt{\sin (x) \cos ^3(x)} \left ((4+4 i) \Pi \left (-i;\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{x}{2}\right )}\right )\right |-1\right )-(4+4 i) \Pi \left (i;\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{x}{2}\right )}\right )\right |-1\right )+\sqrt [4]{-1} \left (-\Pi \left (-\sqrt [4]{-1};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{x}{2}\right )}\right )\right |-1\right )+\Pi \left (\sqrt [4]{-1};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{x}{2}\right )}\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{x}{2}\right )}\right )\right |-1\right )+\Pi \left ((-1)^{3/4};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{x}{2}\right )}\right )\right |-1\right )\right )\right )}{\sqrt{\tan \left (\frac{x}{2}\right )} \left (\tan ^2\left (\frac{x}{2}\right )-1\right ) \sqrt{\cos (x) \sec ^2\left (\frac{x}{2}\right )}}-\frac{\cos (x) \csc \left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \sqrt{\sin ^3(x) \cos (x)} \left (-\log \left (\tan ^4\left (\frac{x}{2}\right )+1\right )-2 \log \left (\tan \left (\frac{x}{2}\right )\right )+4 \log \left (\sec ^2\left (\frac{x}{2}\right )\right )\right )}{8 \sqrt{\sin (x) \cos ^3(x)}}+\frac{\sqrt{2 \sin (2 x)-\sin (4 x)} \sqrt{\tan (x)} \left (\tan ^2(x)+2\right ) \csc ^2(x) \sec ^2(x) \left (4 \log \left (\sqrt{\tan (x)}\right )-\log \left (\tan ^2(x)+2\right )\right )}{4 \sqrt{2} (\cos (2 x)+3) \left (\tan ^2(x)+1\right )^2} \]

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]

-(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[Sqrt[Tan[x/2]]], -1] - (4 + 4*I)*Ellipt
icPi[I, -ArcSin[Sqrt[Tan[x/2]]], -1] + (-1)^(1/4)*(-EllipticPi[-(-1)^(1/4), -ArcSin[Sqrt[Tan[x/2]]], -1] + Ell
ipticPi[(-1)^(1/4), -ArcSin[Sqrt[Tan[x/2]]], -1] - EllipticPi[-(-1)^(3/4), -ArcSin[Sqrt[Tan[x/2]]], -1] + Elli
pticPi[(-1)^(3/4), -ArcSin[Sqrt[Tan[x/2]]], -1]))*Sec[x/2]^4*Sqrt[Cos[x]^3*Sin[x]])/(Sqrt[Cos[x]*Sec[x/2]^2]*S
qrt[Tan[x/2]]*(-1 + Tan[x/2]^2)) + 4/Sqrt[Tan[x]] - (2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[x
]^2/(2*(1 - Sin[x]^2/2))]*(2 - Sin[x]^2)*Tan[x]^(3/2))/(3*(1 - Sin[x]^2/2)^(3/4)*(-2 + Sin[x]^2)) + Sqrt[2*Sin
[2*x] + Sin[4*x]]*(Sqrt[2]*Cot[x] + Sqrt[2]*Tan[x]) + (Csc[x]^2*(4*Log[Sqrt[Tan[x]]] - Log[2 + Tan[x]^2])*Sec[
x]^2*Sqrt[2*Sin[2*x] - Sin[4*x]]*Sqrt[Tan[x]]*(2 + Tan[x]^2))/(4*Sqrt[2]*(3 + Cos[2*x])*(1 + Tan[x]^2)^2)

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Maple [C]  time = 3.754, size = 23475, normalized size = 64.5 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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) - 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) - sq
rt(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)*c
os(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), -co
s(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) + s
qrt(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)))*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)*co
s(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*arctan
2(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)*si
n(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*arc
tan2(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*s
qrt(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)*co
s(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))*co
s(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) + sqr
t(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)))*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) - 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)*si
n(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))*s
in(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) + 1/2*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 - 2*
cos(x) + 1)

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

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: UnboundLocalError

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Sympy [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((-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., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{\cos \left (x\right ) \sin \left (x\right )^{3}} - 2 \, \sin \left (2 \, x\right )}{\sqrt{\cos \left (x\right )^{3} \sin \left (x\right )} - \sqrt{\tan \left (x\right )}}\,{d x} \end{align*}

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)