\(\int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx\) [416]

Optimal result

Integrand size = 27, antiderivative size = 108 \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=-\sqrt {2} \log \left (\cos (x)+\sin (x)-\sqrt {2} \sec (x) \sqrt {\cos ^3(x) \sin (x)}\right )-\frac {\arcsin (\cos (x)-\sin (x)) \cos (x) \sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}}-\frac {\text {arctanh}(\sin (x)) \cos (x) \sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}}-\frac {\sin (2 x)}{\sqrt {\cos ^3(x) \sin (x)}} \]

[Out]

-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)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(108)=216\).

Time = 1.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.17, number of steps used = 27, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6851, 6857, 221, 335, 217, 1179, 642, 1176, 631, 210, 327} \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=-\sqrt {2} \cot (x) \sec ^2(x)^{3/2} \text {arcsinh}(\tan (x)) \sqrt {\sin (x) \cos (x)} \sqrt {\sin (x) \cos ^3(x)}-\frac {\sqrt {2} \sec ^2(x) \arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \sec ^2(x) \arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {\tan (x)}}-2 \sec ^2(x) \sqrt {\sin (x) \cos ^3(x)}-\frac {\sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin (x) \cos ^3(x)}}{\sqrt {2} \sqrt {\tan (x)}} \]

[In]

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

[Out]

-2*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]] - Sqrt[2]*ArcSinh[Tan[x]]*Cot[x]*(Sec[x]^2)^(3/2)*Sqrt[Cos[x]*Sin[x]]*Sqrt[C
os[x]^3*Sin[x]] - (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[x]]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/Sqrt[Tan[x]] + (Sq
rt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[x]]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/Sqrt[Tan[x]] - (Log[1 - Sqrt[2]*Sqrt[Ta
n[x]] + Tan[x]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/(Sqrt[2]*Sqrt[Tan[x]]) + (Log[1 + Sqrt[2]*Sqrt[Tan[x]] + Tan[x
]]*Sec[x]^2*Sqrt[Cos[x]^3*Sin[x]])/(Sqrt[2]*Sqrt[Tan[x]])

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 221

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

Rule 327

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {\frac {x}{\left (1+x^2\right )^2}} \left (1-x^2-\frac {x}{\sqrt {\frac {x}{2+2 x^2}}}\right )}{x} \, dx,x,\tan (x)\right ) \\ & = \frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1-x^2-\frac {x}{\sqrt {\frac {x}{2+2 x^2}}}}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}} \\ & = \frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {2} \sqrt {\frac {x}{1+x^2}}}{\sqrt {x}}+\frac {1}{\sqrt {x} \left (1+x^2\right )}-\frac {x^{3/2}}{1+x^2}\right ) \, dx,x,\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 {x} \left (1+x^2\right )} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}}-\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{1+x^2} \, dx,x,\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 {\sqrt {\frac {x}{1+x^2}}}{\sqrt {x}} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)}} \\ & = -2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\left (\sqrt {2} \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tan (x)\right )+\frac {\left (\sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \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 {1}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}} \\ & = -2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \text {arcsinh}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}+\frac {\left (\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 (\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}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{\sqrt {\tan (x)}} \\ & = -2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \text {arcsinh}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (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 )}{2 \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 )}{2 \sqrt {\tan (x)}}+\frac {\left (\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 (\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 (\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 )}{2 \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 )}{2 \sqrt {2} \sqrt {\tan (x)}} \\ & = -2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \text {arcsinh}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{2 \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)}}{2 \sqrt {2} \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 )}{2 \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 )}{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 )}{2 \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 )}{2 \sqrt {2} \sqrt {\tan (x)}}+\frac {\left (\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 {2} \sqrt {\tan (x)}}-\frac {\left (\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 {2} \sqrt {\tan (x)}} \\ & = -2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \text {arcsinh}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {2} \sqrt {\tan (x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\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 {\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 (\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 {2} \sqrt {\tan (x)}}-\frac {\left (\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 {2} \sqrt {\tan (x)}} \\ & = -2 \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}-\sqrt {2} \text {arcsinh}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt {\cos (x) \sin (x)} \sqrt {\cos ^3(x) \sin (x)}-\frac {\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\cos ^3(x) \sin (x)}}{\sqrt {\tan (x)}}+\frac {\sqrt {2} \arctan \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)}} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\frac {-4 \cos ^3(x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\cos ^2(x)\right ) \sin (x)-3 \cos (x) \sqrt [4]{\sin ^2(x)} \left (2 \sin (x)+\text {arctanh}(\sin (x)) \sqrt {\sin (2 x)}\right )}{3 \sqrt {\cos ^3(x) \sin (x)} \sqrt [4]{\sin ^2(x)}} \]

[In]

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

[Out]

(-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))

Maple [B] (warning: unable to verify)

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	\(-\frac {2 \sin \left (x \right ) \cos \left (x \right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {2 \sqrt {2}\, \cos \left (x \right ) \sqrt {\cos \left (x \right ) \sin \left (x \right )}\, \operatorname {arctanh}\left (-\csc \left (x \right )+\cot \left (x \right )\right )}{\sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}+\frac {\sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2+2 \cot \left (x \right )\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )-\cos \left (x \right )+1}{-1+\cos \left (x \right )}\right )-\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2+2 \cot \left (x \right )\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )\right ) \left (\cos ^{2}\left (x \right )+\cos \left (x \right )\right ) \sqrt {2}}{2 \sqrt {\left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )}}\)	\(247\)

[In]

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

[Out]

-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)*sin(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)

Fricas [C] (verification not implemented)

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=\frac {-\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {2 \, \cos \left (x\right )^{3} + 2 i \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {2 \, \cos \left (x\right )^{3} + 2 i \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (-\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {2 \, \cos \left (x\right )^{3} - 2 i \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (-\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) - \left (i + 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {2 \, \cos \left (x\right )^{3} - 2 i \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) - \left (i + 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (-\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) - \left (i - 1\right ) \, \sqrt {2} \cos \left (x\right )^{2} \log \left (\frac {\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} {\left (-\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} - \cos \left (x\right )}{\cos \left (x\right )}\right ) + 4 \, \sqrt {2} \cos \left (x\right )^{2} \log \left (-\frac {\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )}}{\cos \left (x\right )^{4}}\right ) - 16 \, \sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}}{8 \, \cos \left (x\right )^{2}} \]

[In]

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

[Out]

1/8*(-(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*cos(x)^2*s
in(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)*s
qrt(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*cos(x)^2*sin(x) + sqrt(c
os(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((sqrt(cos(x)^3*sin(x))*(-(I + 1)*sqrt(2)*cos(x) + (I - 1)*sqrt(2)*si
n(x)) - cos(x))/cos(x)) + 4*sqrt(2)*cos(x)^2*log(-(cos(x)^4 - 2*cos(x)^2 + 2*sqrt(cos(x)^3*sin(x))*sqrt(cos(x)
*sin(x)))/cos(x)^4) - 16*sqrt(cos(x)^3*sin(x)))/cos(x)^2

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int { -\frac {\sqrt {\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}} \,d x } \]

[In]

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

[Out]

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(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(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*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*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), -co
s(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)))*s
in(1/2*arctan2(sin(x), cos(x) + 1)))*cos(3/2*arctan2(sin(2*x), cos(2*x) + 1)) - (((cos(4*x) + 1)*cos(1/2*arcta
n2(sin(x), -cos(x) + 1)) - sin(4*x)*sin(1/2*arctan2(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(sin(x), -cos(x) + 1)))*si
n(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*co
s(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)*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + 1/2*sqrt(2)*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)

Giac [F]

\[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int { -\frac {\sqrt {\sin \left (2 \, x\right )} - \cos \left (2 \, x\right )}{\sqrt {\cos \left (x\right )^{3} \sin \left (x\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (2 x)-\sqrt {\sin (2 x)}}{\sqrt {\cos ^3(x) \sin (x)}} \, dx=\int \frac {\cos \left (2\,x\right )-\sqrt {\sin \left (2\,x\right )}}{\sqrt {{\cos \left (x\right )}^3\,\sin \left (x\right )}} \,d x \]

[In]

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

[Out]

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