Integrand size = 27, antiderivative size = 108 \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=-\sqrt {2} \log \left (\cos (x)+\sin (x)-\sqrt {2} \sec (x) \sqrt {\cos ^3(x) \sin (x)}\right )-\frac {\arcsin (\cos (x)-\sin (x)) \cos (x) \sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}}-\frac {\text {arctanh}(\sin (x)) \cos (x) \sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}}-\frac {\sin (2 x)}{\sqrt {\cos ^3(x) \sin (x)}} \]
-ln(cos(x)+sin(x)-sec(x)*2^(1/2)*(cos(x)^3*sin(x))^(1/2))*2^(1/2)-sin(2*x) /(cos(x)^3*sin(x))^(1/2)-arcsin(cos(x)-sin(x))*cos(x)*sin(2*x)^(1/2)/(cos( x)^3*sin(x))^(1/2)-arctanh(sin(x))*cos(x)*sin(2*x)^(1/2)/(cos(x)^3*sin(x)) ^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.50 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\frac {-4 \cos ^3(x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\cos ^2(x)\right ) \sin (x)-3 \cos (x) \sqrt [4]{\sin ^2(x)} \left (2 \sin (x)+\text {arctanh}(\sin (x)) \sqrt {\sin (2 x)}\right )}{3 \sqrt {\cos ^3(x) \sin (x)} \sqrt [4]{\sin ^2(x)}} \]
(-4*Cos[x]^3*Hypergeometric2F1[3/4, 3/4, 7/4, Cos[x]^2]*Sin[x] - 3*Cos[x]* (Sin[x]^2)^(1/4)*(2*Sin[x] + ArcTanh[Sin[x]]*Sqrt[Sin[2*x]]))/(3*Sqrt[Cos[ x]^3*Sin[x]]*(Sin[x]^2)^(1/4))
Time = 1.16 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.59, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 4889, 7270, 2035, 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.
| \(\displaystyle \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\sin (x) \cos ^3(x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\sin (x) \cos (x)^3}}dx\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \int \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (-\tan ^2(x)-\frac {\sqrt {2} \tan (x)}{\sqrt {\frac {\tan (x)}{\tan ^2(x)+1}}}+1\right ) \cot (x)d\tan (x)\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \int \frac {-\tan ^2(x)-\frac {\sqrt {2} \tan (x)}{\sqrt {\frac {\tan (x)}{\tan ^2(x)+1}}}+1}{\sqrt {\tan (x)} \left (\tan ^2(x)+1\right )}d\tan (x)}{\sqrt {\tan (x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \int \frac {-\tan ^2(x)-\frac {\sqrt {2} \tan (x)}{\sqrt {\frac {\tan (x)}{\tan ^2(x)+1}}}+1}{\tan ^2(x)+1}d\sqrt {\tan (x)}}{\sqrt {\tan (x)}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \int \left (-\frac {\tan ^2(x)}{\tan ^2(x)+1}-\sqrt {2} \sqrt {\frac {\tan (x)}{\tan ^2(x)+1}}+\frac {1}{\tan ^2(x)+1}\right )d\sqrt {\tan (x)}}{\sqrt {\tan (x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^2}} \left (\tan ^2(x)+1\right ) \left (-\frac {\sqrt {\frac {\tan (x)}{\tan ^2(x)+1}} \sqrt {\tan ^2(x)+1} \cot (x) \text {arcsinh}(\tan (x))}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}-\sqrt {\tan (x)}-\frac {\log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}\right )}{\sqrt {\tan (x)}}\) |
(2*Sqrt[Tan[x]/(1 + Tan[x]^2)^2]*(1 + Tan[x]^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt [Tan[x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]]/Sqrt[2] - Log[1 - Sq rt[2]*Sqrt[Tan[x]] + Tan[x]]/(2*Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x]]/(2*Sqrt[2]) - Sqrt[Tan[x]] - (ArcSinh[Tan[x]]*Cot[x]*Sqrt[Tan[x]/( 1 + Tan[x]^2)]*Sqrt[1 + Tan[x]^2])/Sqrt[2]))/Sqrt[Tan[x]]
3.5.16.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(92)=184\).
Time = 3.68 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.29
| method | result | size |
| default | \(-\frac {2 \sin \left (x \right ) \cos \left (x \right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {2 \sqrt {2}\, \cos \left (x \right ) \sqrt {\cos \left (x \right ) \sin \left (x \right )}\, \operatorname {arctanh}\left (-\csc \left (x \right )+\cot \left (x \right )\right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {\sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2+2 \cot \left (x \right )\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )-\cos \left (x \right )+1}{-1+\cos \left (x \right )}\right )-\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2+2 \cot \left (x \right )\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )\right ) \left (\cos ^{2}\left (x \right )+\cos \left (x \right )\right ) \sqrt {2}}{2 \sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}\) | \(247\) |
| parts | \(-\frac {2 \sin \left (x \right ) \cos \left (x \right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {2 \sqrt {2}\, \cos \left (x \right ) \sqrt {\cos \left (x \right ) \sin \left (x \right )}\, \operatorname {arctanh}\left (-\csc \left (x \right )+\cot \left (x \right )\right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {\sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2+2 \cot \left (x \right )\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )-\cos \left (x \right )+1}{-1+\cos \left (x \right )}\right )-\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2+2 \cot \left (x \right )\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )\right ) \left (\cos ^{2}\left (x \right )+\cos \left (x \right )\right ) \sqrt {2}}{2 \sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}\) | \(247\) |
-2*sin(x)*cos(x)/(cos(x)^3*sin(x))^(1/2)+2*2^(1/2)*cos(x)*(cos(x)*sin(x))^ (1/2)*arctanh(-csc(x)+cot(x))/(cos(x)^3*sin(x))^(1/2)+1/2*(cos(x)*sin(x)/( cos(x)+1)^2)^(1/2)*(ln(2*cot(x)*2^(1/2)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2) +2*csc(x)*2^(1/2)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)+2+2*cot(x))+2*arctan( (2^(1/2)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)*sin(x)-cos(x)+1)/(-1+cos(x)))- ln(-2*cot(x)*2^(1/2)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)-2*csc(x)*2^(1/2)*( cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)+2+2*cot(x))+2*arctan((2^(1/2)*(cos(x)*si n(x)/(cos(x)+1)^2)^(1/2)*sin(x)+cos(x)-1)/(-1+cos(x))))/(cos(x)^3*sin(x))^ (1/2)*(cos(x)^2+cos(x))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 479, normalized size of antiderivative = 4.44 \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx =\text {Too large to display} \]
1/8*(-(I - 1)*sqrt(2)*cos(x)^2*log((2*cos(x)^3 + 2*I*cos(x)^2*sin(x) + sqr t(cos(x)^3*sin(x))*((I + 1)*sqrt(2)*cos(x) + (I - 1)*sqrt(2)*sin(x)) - cos (x))/cos(x)) + (I - 1)*sqrt(2)*cos(x)^2*log((2*cos(x)^3 + 2*I*cos(x)^2*sin (x) + sqrt(cos(x)^3*sin(x))*(-(I + 1)*sqrt(2)*cos(x) - (I - 1)*sqrt(2)*sin (x)) - cos(x))/cos(x)) + (I + 1)*sqrt(2)*cos(x)^2*log((2*cos(x)^3 - 2*I*co s(x)^2*sin(x) + sqrt(cos(x)^3*sin(x))*(-(I - 1)*sqrt(2)*cos(x) - (I + 1)*s qrt(2)*sin(x)) - cos(x))/cos(x)) - (I + 1)*sqrt(2)*cos(x)^2*log((2*cos(x)^ 3 - 2*I*cos(x)^2*sin(x) + sqrt(cos(x)^3*sin(x))*((I - 1)*sqrt(2)*cos(x) + (I + 1)*sqrt(2)*sin(x)) - cos(x))/cos(x)) + (I - 1)*sqrt(2)*cos(x)^2*log(( sqrt(cos(x)^3*sin(x))*((I + 1)*sqrt(2)*cos(x) - (I - 1)*sqrt(2)*sin(x)) - cos(x))/cos(x)) - (I + 1)*sqrt(2)*cos(x)^2*log((sqrt(cos(x)^3*sin(x))*(-(I - 1)*sqrt(2)*cos(x) + (I + 1)*sqrt(2)*sin(x)) - cos(x))/cos(x)) + (I + 1) *sqrt(2)*cos(x)^2*log((sqrt(cos(x)^3*sin(x))*((I - 1)*sqrt(2)*cos(x) - (I + 1)*sqrt(2)*sin(x)) - cos(x))/cos(x)) - (I - 1)*sqrt(2)*cos(x)^2*log((sqr t(cos(x)^3*sin(x))*(-(I + 1)*sqrt(2)*cos(x) + (I - 1)*sqrt(2)*sin(x)) - co s(x))/cos(x)) + 4*sqrt(2)*cos(x)^2*log(-(cos(x)^4 - 2*cos(x)^2 + 2*sqrt(co s(x)^3*sin(x))*sqrt(cos(x)*sin(x)))/cos(x)^4) - 16*sqrt(cos(x)^3*sin(x)))/ cos(x)^2
Timed out. \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int { -\frac {\sqrt {\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}} \,d x } \]
1/2*sqrt(2)*integrate(2*((((cos(4*x) + 1)*cos(1/2*arctan2(sin(x), -cos(x) + 1)) - sin(4*x)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*cos(1/2*arctan2(si n(x), cos(x) + 1)) + (cos(1/2*arctan2(sin(x), -cos(x) + 1))*sin(4*x) + (co s(4*x) + 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1)))*cos(3/2*arctan2(sin(2*x), cos(2*x) + 1)) + ((cos(1/2*arctan 2(sin(x), -cos(x) + 1))*sin(4*x) + (cos(4*x) + 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*cos(1/2*arctan2(sin(x), cos(x) + 1)) - ((cos(4*x) + 1)*cos( 1/2*arctan2(sin(x), -cos(x) + 1)) - sin(4*x)*sin(1/2*arctan2(sin(x), -cos( x) + 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1)))*sin(3/2*arctan2(sin(2*x), cos(2*x) + 1)))/((cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1)^(3/4)*(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*sqrt(2)*integrate(-2*(((cos(1/2*arctan2(sin(x), -cos(x) + 1 ))*sin(4*x) + (cos(4*x) + 1)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*cos(1/ 2*arctan2(sin(x), cos(x) + 1)) - ((cos(4*x) + 1)*cos(1/2*arctan2(sin(x), - cos(x) + 1)) - sin(4*x)*sin(1/2*arctan2(sin(x), -cos(x) + 1)))*sin(1/2*arc tan2(sin(x), cos(x) + 1)))*cos(3/2*arctan2(sin(2*x), cos(2*x) + 1)) - (((c os(4*x) + 1)*cos(1/2*arctan2(sin(x), -cos(x) + 1)) - sin(4*x)*sin(1/2*arct an2(sin(x), -cos(x) + 1)))*cos(1/2*arctan2(sin(x), cos(x) + 1)) + (cos(1/2 *arctan2(sin(x), -cos(x) + 1))*sin(4*x) + (cos(4*x) + 1)*sin(1/2*arctan2(s in(x), -cos(x) + 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1)))*sin(3/2*arc...
\[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int { -\frac {\sqrt {\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}} \,d x } \]
Timed out. \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int \frac {\cos \left (2\,x\right )-\sqrt {\sin \left (2\,x\right )}}{\sqrt {{\cos \left (x\right )}^3\,\sin \left (x\right )}} \,d x \]
\[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {\cos \left (x \right )}\, \cos \left (2 x \right )}{{| \cos \left (x \right )|} \cos \left (x \right ) \sin \left (x \right )}d x -\left (\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {\sin \left (2 x \right )}\, \sqrt {\cos \left (x \right )}}{{| \cos \left (x \right )|} \cos \left (x \right ) \sin \left (x \right )}d x \right ) \]