Integrand size = 41, antiderivative size = 364 \[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=-2 \sqrt {2} \coth ^{-1}\left (\frac {\cos (x) (\cos (x)+\sin (x))}{\sqrt {2} \sqrt {\cos ^3(x) \sin (x)}}\right )+\sqrt [4]{2} \coth ^{-1}\left (\frac {\cos (x) \left (\sqrt {2} \cos (x)+\sin (x)\right )}{2^{3/4} \sqrt {\cos ^3(x) \sin (x)}}\right )-\sqrt [4]{2} \coth ^{-1}\left (\frac {\sqrt {2}+\tan (x)}{2^{3/4} \sqrt {\tan (x)}}\right )-2 \sqrt {2} \arctan \left (\frac {\cos (x) (\cos (x)-\sin (x))}{\sqrt {2} \sqrt {\cos ^3(x) \sin (x)}}\right )+\sqrt [4]{2} \arctan \left (\frac {\cos (x) \left (\sqrt {2} \cos (x)-\sin (x)\right )}{2^{3/4} \sqrt {\cos ^3(x) \sin (x)}}\right )-\sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\tan (x)}{2^{3/4} \sqrt {\tan (x)}}\right )+4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {1}{4} \csc ^2(x) \log \left (1+\cos ^2(x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {1}{2} \csc ^2(x) \log (\sin (x)) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)} \sqrt {\cos (x) \sin ^3(x)}+\frac {4}{\sqrt {\tan (x)}}-\frac {1}{4} \csc ^2(x) \log \left (1+\cos ^2(x)\right ) \sqrt {\cos (x) \sin ^3(x)} \sqrt {\tan (x)}+\frac {1}{2} \csc ^2(x) \log (\sin (x)) \sqrt {\cos (x) \sin ^3(x)} \sqrt {\tan (x)} \]
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*arcta n(1/2*cos(x)*(cos(x)-sin(x))*2^(1/2)/(cos(x)^3*sin(x))^(1/2))*2^(1/2)+4*cs c(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)*sin(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*c sc(x)^2*ln(1+cos(x)^2)*(cos(x)*sin(x)^3)^(1/2)*tan(x)^(1/2)+1/2*csc(x)^2*l n(sin(x))*(cos(x)*sin(x)^3)^(1/2)*tan(x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 19.66 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=4 \csc (x) \sec (x) \sqrt {\cos ^3(x) \sin (x)}+\frac {\cot (x) \left (-2 \log \left (\sec ^2(x)\right )+2 \log (\tan (x))+\log \left (2+\tan ^2(x)\right )\right ) \sqrt {\cos (x) \sin ^3(x)}}{4 \sqrt {\cos ^3(x) \sin (x)}}+\frac {4}{\sqrt {\tan (x)}}+\frac {\left (2 \arctan \left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )-2 \arctan \left (1+\sqrt [4]{2} \sqrt {\tan (x)}\right )-4 \sqrt [4]{2} \arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )+4 \sqrt [4]{2} \arctan \left (1+\sqrt {2} \sqrt {\tan (x)}\right )+2 \sqrt [4]{2} \log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )-2 \sqrt [4]{2} \log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )-\log \left (2-2 \sqrt [4]{2} \sqrt {\tan (x)}+\sqrt {2} \tan (x)\right )+\log \left (2+2 \sqrt [4]{2} \sqrt {\tan (x)}+\sqrt {2} \tan (x)\right )\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2^{3/4} \sqrt {\tan (x)}}+\frac {1}{4} \csc ^2(x) \left (2 \log (\tan (x))-\log \left (2+\tan ^2(x)\right )\right ) \sqrt {\cos (x) \sin ^3(x)} \sqrt {\tan (x)}+\frac {4 \sqrt {2} \cos ^2(x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\frac {2 \sin ^2(x)}{3+\cos (2 x)}\right ) \tan ^{\frac {3}{2}}(x)}{3 (3+\cos (2 x))^{3/4}} \]
4*Csc[x]*Sec[x]*Sqrt[Cos[x]^3*Sin[x]] + (Cot[x]*(-2*Log[Sec[x]^2] + 2*Log[ Tan[x]] + Log[2 + Tan[x]^2])*Sqrt[Cos[x]*Sin[x]^3])/(4*Sqrt[Cos[x]^3*Sin[x ]]) + 4/Sqrt[Tan[x]] + ((2*ArcTan[1 - 2^(1/4)*Sqrt[Tan[x]]] - 2*ArcTan[1 + 2^(1/4)*Sqrt[Tan[x]]] - 4*2^(1/4)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[x]]] + 4*2^ (1/4)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]] + 2*2^(1/4)*Log[1 - Sqrt[2]*Sqrt[Ta n[x]] + Tan[x]] - 2*2^(1/4)*Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x]] - Log[2 - 2*2^(1/4)*Sqrt[Tan[x]] + Sqrt[2]*Tan[x]] + Log[2 + 2*2^(1/4)*Sqrt[Tan[x ]] + Sqrt[2]*Tan[x]])*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/(2^(3/4)*Sqrt[Tan[x] ]) + (Csc[x]^2*(2*Log[Tan[x]] - Log[2 + Tan[x]^2])*Sqrt[Cos[x]*Sin[x]^3]*S qrt[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))
Leaf count is larger than twice the leaf count of optimal. \(781\) vs. \(2(364)=728\).
Time = 3.30 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.15, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {3042, 4889, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {\sin ^3(x) \cos (x)}-2 \sin (2 x)}{\sqrt {\tan (x)}-\sqrt {\sin (x) \cos ^3(x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (x)^3 \cos (x)}-2 \sin (2 x)}{\sqrt {\tan (x)}-\sqrt {\sin (x) \cos (x)^3}}dx\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \int \frac {\sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}}-\frac {4 \tan (x)}{\tan ^2(x)+1}}{\left (\tan ^2(x)+1\right ) \left (\sqrt {\tan (x)}-\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}}\right )}d\tan (x)\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {4 \tan (x)}{\left (\tan ^2(x)+1\right )^2 \left (\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}}-\sqrt {\tan (x)}\right )}-\frac {\sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}}}{\left (\tan ^2(x)+1\right ) \left (\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}}-\sqrt {\tan (x)}\right )}\right )d\tan (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \arctan \left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}-\frac {\sqrt [4]{2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \arctan \left (\sqrt [4]{2} \sqrt {\tan (x)}+1\right )}{\sqrt {\tan (x)}}-\frac {2 \sqrt {2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}}+\frac {2 \sqrt {2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {\tan (x)}}-\sqrt [4]{2} \arctan \left (1-\sqrt [4]{2} \sqrt {\tan (x)}\right )+\sqrt [4]{2} \arctan \left (\sqrt [4]{2} \sqrt {\tan (x)}+1\right )+\frac {4}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {\tan (x)}}-\frac {\sqrt {2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {\tan (x)}}-\frac {\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \log \left (\tan (x)-2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right )}{2^{3/4} \sqrt {\tan (x)}}+\frac {\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \log \left (\tan (x)+2^{3/4} \sqrt {\tan (x)}+\sqrt {2}\right )}{2^{3/4} \sqrt {\tan (x)}}+\frac {\sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \log (\tan (x))}{2 \tan ^{\frac {3}{2}}(x)}-\frac {\sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \log \left (\tan ^2(x)+2\right )}{4 \tan ^{\frac {3}{2}}(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 \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \cot (x)+\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right )^2 \cot ^2(x) \log \left (\sqrt {\tan (x)}\right )-\frac {1}{2} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right )^2 \cot ^2(x) \log \left (\tan ^2(x)+1\right )+\frac {1}{4} \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \sqrt {\frac {\tan ^3(x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right )^2 \cot ^2(x) \log \left (\tan ^2(x)+2\right )\) |
-(2^(1/4)*ArcTan[1 - 2^(1/4)*Sqrt[Tan[x]]]) + 2^(1/4)*ArcTan[1 + 2^(1/4)*S qrt[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/Sqrt[Tan[x]] + 4*Cot[ x]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2) + (2^(1/4)*ArcTan[1 - 2^(1 /4)*Sqrt[Tan[x]]]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2))/Sqrt[Tan[x ]] - (2^(1/4)*ArcTan[1 + 2^(1/4)*Sqrt[Tan[x]]]*Sqrt[Tan[x]/(1 + Tan[x]^2)^ 2]*(1 + Tan[x]^2))/Sqrt[Tan[x]] - (2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[x ]]]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2))/Sqrt[Tan[x]] + (2*Sqrt[2 ]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[ x]^2))/Sqrt[Tan[x]] + (Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]*Sqrt [Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2))/Sqrt[Tan[x]] - (Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2 ))/Sqrt[Tan[x]] - (Log[Sqrt[2] - 2^(3/4)*Sqrt[Tan[x]] + Tan[x]]*Sqrt[Tan[x ]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2))/(2^(3/4)*Sqrt[Tan[x]]) + (Log[Sqrt[2] + 2^(3/4)*Sqrt[Tan[x]] + Tan[x]]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x] ^2))/(2^(3/4)*Sqrt[Tan[x]]) + (Log[Tan[x]]*Sqrt[Tan[x]^3/(1 + Tan[x]^2)^2] *(1 + Tan[x]^2))/(2*Tan[x]^(3/2)) - (Log[2 + Tan[x]^2]*Sqrt[Tan[x]^3/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2))/(4*Tan[x]^(3/2)) + Cot[x]^2*Log[Sqrt[Tan[x]]] *Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*Sqrt[Tan[x]^3/(1 + Tan[x]^2)^2]*(1 + Tan[x] ^2)^2 - (Cot[x]^2*Log[1 + Tan[x]^2]*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*Sqrt[...
3.5.17.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 57.05 (sec) , antiderivative size = 133928, normalized size of antiderivative = 367.93
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(133928\) |
| parts | \(\text {Expression too large to display}\) | \(149826\) |
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)
Exception generated. \[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=\text {Exception raised: TypeError} \]
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")
Timed out. \[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=\text {Timed out} \]
integrate((-2*sin(2*x)+(cos(x)*sin(x)**3)**(1/2))/(-(cos(x)**3*sin(x))**(1 /2)+tan(x)**(1/2)),x)
\[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=\int { -\frac {\sqrt {\cos \left (x\right ) \sin \left (x\right )^{3}} - 2 \, \sin \left (2 \, x\right )}{\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} - \sqrt {\tan \left (x\right )}} \,d x } \]
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")
-2*integrate(-1/4*(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^(1/4)*(((((sq rt(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) - sq rt(2)*sin(x))*sin(4*x) - sqrt(2)*cos(3*x) - 2*sqrt(2)*cos(2*x) - sqrt(2)*c os(x) - sqrt(2)*sin(3*x) - 2*sqrt(2)*sin(2*x) - sqrt(2)*sin(x))*cos(1/2*ar ctan2(sin(x), -cos(x) + 1)) - ((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 (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*arc tan2(sin(x), cos(x) + 1)) + (((sqrt(2)*cos(3*x) + 2*sqrt(2)*cos(2*x) + sqr t(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)*s in(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)*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(4*x) - (sqrt(2)*cos(3*x) + 2*sqrt(2)*...
\[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=\int { -\frac {\sqrt {\cos \left (x\right ) \sin \left (x\right )^{3}} - 2 \, \sin \left (2 \, x\right )}{\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} - \sqrt {\tan \left (x\right )}} \,d x } \]
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")
Timed out. \[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=-\int \frac {2\,\sin \left (2\,x\right )-\sqrt {\cos \left (x\right )\,{\sin \left (x\right )}^3}}{\sqrt {\mathrm {tan}\left (x\right )}-\sqrt {{\cos \left (x\right )}^3\,\sin \left (x\right )}} \,d x \]
\[ \int \frac {\sqrt {\cos (x) \sin ^3(x)}-2 \sin (2 x)}{-\sqrt {\cos ^3(x) \sin (x)}+\sqrt {\tan (x)}} \, dx=\int \frac {-\sqrt {\cos \left (x \right ) \sin \left (x \right )}\, {| \sin \left (x \right )|}+2 \sin \left (2 x \right )}{\sqrt {\cos \left (x \right ) \sin \left (x \right )}\, {| \cos \left (x \right )|}-\sqrt {\tan \left (x \right )}}d x \]
int(( - sqrt(cos(x)*sin(x))*abs(sin(x)) + 2*sin(2*x))/(sqrt(cos(x)*sin(x)) *abs(cos(x)) - sqrt(tan(x))),x)