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

Optimal. Leaf size=108 \[ -\frac{\sqrt{\sin (2 x)} \cos (x) \sin ^{-1}(\cos (x)-\sin (x))}{\sqrt{\sin (x) \cos ^3(x)}}-\frac{\sin (2 x)}{\sqrt{\sin (x) \cos ^3(x)}}-\frac{\sqrt{\sin (2 x)} \cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{\sin (x) \cos ^3(x)}}-\sqrt{2} \log \left (\sin (x)+\cos (x)-\sqrt{2} \sec (x) \sqrt{\sin (x) \cos ^3(x)}\right ) \]

[Out]

-(Sqrt[2]*Log[Cos[x] + Sin[x] - Sqrt[2]*Sec[x]*Sqrt[Cos[x]^3*Sin[x]]]) - (ArcSin[Cos[x] - Sin[x]]*Cos[x]*Sqrt[
Sin[2*x]])/Sqrt[Cos[x]^3*Sin[x]] - (ArcTanh[Sin[x]]*Cos[x]*Sqrt[Sin[2*x]])/Sqrt[Cos[x]^3*Sin[x]] - Sin[2*x]/Sq
rt[Cos[x]^3*Sin[x]]

________________________________________________________________________________________

Rubi [B]  time = 1.53986, antiderivative size = 234, normalized size of antiderivative = 2.17, number of steps used = 27, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {6719, 6725, 215, 329, 211, 1165, 628, 1162, 617, 204, 321} \[ -2 \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}-\frac{\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}+\frac{\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (x)}+1\right ) \sec ^2(x) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{\tan (x)}}-\frac{\sec ^2(x) \log \left (\tan (x)-\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{2} \sqrt{\tan (x)}}+\frac{\sec ^2(x) \log \left (\tan (x)+\sqrt{2} \sqrt{\tan (x)}+1\right ) \sqrt{\sin (x) \cos ^3(x)}}{\sqrt{2} \sqrt{\tan (x)}}-\sqrt{2} \cot (x) \sec ^2(x)^{3/2} \sqrt{\sin (x) \cos (x)} \sqrt{\sin (x) \cos ^3(x)} \sinh ^{-1}(\tan (x)) \]

Antiderivative was successfully verified.

[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 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 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 215

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 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 211

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 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 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 321

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]

Rubi steps

\begin{align*} \int \frac{\cos (2 x)-\sqrt{\sin (2 x)}}{\sqrt{\cos ^3(x) \sin (x)}} \, dx &=\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \sinh ^{-1}(\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 ) \operatorname{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 ) \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}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )}{\sqrt{\tan (x)}}\\ &=-2 \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}-\sqrt{2} \sinh ^{-1}(\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \sinh ^{-1}(\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (x)}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right ) \sec ^2(x) \sqrt{\cos ^3(x) \sin (x)}}{\sqrt{2} \sqrt{\tan (x)}}+\frac{\tan ^{-1}\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 ) \operatorname{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 ) \operatorname{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} \sinh ^{-1}(\tan (x)) \cot (x) \sec ^2(x)^{3/2} \sqrt{\cos (x) \sin (x)} \sqrt{\cos ^3(x) \sin (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{\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)}}\\ \end{align*}

Mathematica [C]  time = 0.261313, size = 105, normalized size = 0.97 \[ \frac{-4 \sin (x) \cos ^3(x) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\cos ^2(x)\right )-3 \sqrt [4]{\sin ^2(x)} \cos (x) \left (2 \sin (x)+\sqrt{\sin (2 x)} \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right )\right )}{3 \sqrt [4]{\sin ^2(x)} \sqrt{\sin (x) \cos ^3(x)}} \]

Antiderivative was successfully verified.

[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] + (-Log[C
os[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sqrt[Sin[2*x]]))/(3*Sqrt[Cos[x]^3*Sin[x]]*(Sin[x]^2)^(1/4))

________________________________________________________________________________________

Maple [C]  time = 0.317, size = 247, normalized size = 2.3 \begin{align*} -2\,{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}}-{\frac{\cos \left ( x \right ) \sqrt{2} \left ( \sin \left ( x \right ) \right ) ^{2}}{\cos \left ( x \right ) -1} \left ( i{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -i{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -2\,{\it EllipticF} \left ( \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) \right ) \sqrt{-{\frac{-\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}}}+2\,{\frac{\cos \left ( x \right ) \sqrt{2}\sqrt{\cos \left ( x \right ) \sin \left ( x \right ) }}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }}{\it Artanh} \left ({\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)*sin(x)/(cos(x)^3*sin(x))^(1/2)-2^(1/2)*(I*EllipticPi((-(-sin(x)-1+cos(x))/sin(x))^(1/2),1/2-1/2*I,1/
2*2^(1/2))-I*EllipticPi((-(-sin(x)-1+cos(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+EllipticPi((-(-sin(x)-1+cos(
x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+EllipticPi((-(-sin(x)-1+cos(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-
2*EllipticF((-(-sin(x)-1+cos(x))/sin(x))^(1/2),1/2*2^(1/2)))*cos(x)*sin(x)^2*(-(-sin(x)-1+cos(x))/sin(x))^(1/2
)*((sin(x)-1+cos(x))/sin(x))^(1/2)*((cos(x)-1)/sin(x))^(1/2)/(cos(x)-1)/(cos(x)^3*sin(x))^(1/2)+2*2^(1/2)*arct
anh((cos(x)-1)/sin(x))*cos(x)*(cos(x)*sin(x))^(1/2)/(cos(x)^3*sin(x))^(1/2)

________________________________________________________________________________________

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((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)

________________________________________________________________________________________

Fricas [B]  time = 76.0529, size = 2007, normalized size = 18.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*arctan(-1/2*(2*cos(x)^4 - 2*cos(x)^3*sin(x) - 2*cos(x)^2 - sqrt(2)*sqrt(cos(x)^3*sin(x))*sqrt(
(4*cos(x)^2*sin(x) + 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x)) - sqrt(2)*sqrt(cos(x)
^3*sin(x)))/(cos(x)^4 + cos(x)^3*sin(x) - cos(x)^2))*cos(x)^2 + 2*sqrt(2)*arctan(1/2*(2*cos(x)^4 - 2*cos(x)^3*
sin(x) - 2*cos(x)^2 + sqrt(2)*sqrt(cos(x)^3*sin(x))*sqrt((4*cos(x)^2*sin(x) - 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*
(cos(x) + sin(x)) + cos(x))/cos(x)) + sqrt(2)*sqrt(cos(x)^3*sin(x)))/(cos(x)^4 + cos(x)^3*sin(x) - cos(x)^2))*
cos(x)^2 - 2*sqrt(2)*arctan(-1/2*(sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) - sin(x)) + (2*cos(x)^2*sin(x) - sqrt(
2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)))*sqrt((4*cos(x)^2*sin(x) + 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x)
+ sin(x)) + cos(x))/cos(x)))/(cos(x)^2*sin(x)))*cos(x)^2 - 2*sqrt(2)*arctan(-1/2*(sqrt(2)*sqrt(cos(x)^3*sin(x)
)*(cos(x) - sin(x)) - (2*cos(x)^2*sin(x) + sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)))*sqrt((4*cos(x)^2*s
in(x) - 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x)))/(cos(x)^2*sin(x)))*cos(x)^2 - sqr
t(2)*cos(x)^2*log((4*cos(x)^2*sin(x) + 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x)) + s
qrt(2)*cos(x)^2*log((4*cos(x)^2*sin(x) - 2*sqrt(2)*sqrt(cos(x)^3*sin(x))*(cos(x) + sin(x)) + cos(x))/cos(x)) -
sqrt(2)*cos(x)^2*log((cos(x)^6 - 8*cos(x)^4 + 4*sqrt(cos(x)^3*sin(x))*(cos(x)^2 - 2)*sqrt(cos(x)*sin(x)) + 8*
cos(x)^2)/cos(x)^6) + 8*sqrt(cos(x)^3*sin(x)))/cos(x)^2

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)