3.634 \(\int \frac{\csc ^2(x) \sec (x)}{\sqrt{\sin (2 x)} (-2+\tan (x))} \, dx\)

Optimal. Leaf size=69 \[ \frac{\cos (x)}{2 \sqrt{\sin (2 x)}}-\frac{5 \sin (x) \tanh ^{-1}\left (\frac{\sqrt{\tan (x)}}{\sqrt{2}}\right )}{2 \sqrt{2} \sqrt{\sin (2 x)} \sqrt{\tan (x)}}+\frac{\cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}} \]

[Out]

Cos[x]/(2*Sqrt[Sin[2*x]]) + (Cos[x]*Cot[x])/(3*Sqrt[Sin[2*x]]) - (5*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x])/(2*S
qrt[2]*Sqrt[Sin[2*x]]*Sqrt[Tan[x]])

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Rubi [A]  time = 0.363889, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4390, 898, 1262, 207} \[ \frac{\cos (x)}{2 \sqrt{\sin (2 x)}}-\frac{5 \sin (x) \tanh ^{-1}\left (\frac{\sqrt{\tan (x)}}{\sqrt{2}}\right )}{2 \sqrt{2} \sqrt{\sin (2 x)} \sqrt{\tan (x)}}+\frac{\cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x]^2*Sec[x])/(Sqrt[Sin[2*x]]*(-2 + Tan[x])),x]

[Out]

Cos[x]/(2*Sqrt[Sin[2*x]]) + (Cos[x]*Cot[x])/(3*Sqrt[Sin[2*x]]) - (5*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x])/(2*S
qrt[2]*Sqrt[Sin[2*x]]*Sqrt[Tan[x]])

Rule 4390

Int[(u_)*((c_.)*sin[v_])^(m_), x_Symbol] :> With[{w = FunctionOfTrig[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]}, D
ist[((c*Sin[v])^m*(c*Tan[v/2])^m)/Sin[v/2]^(2*m), Int[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x], x] /;  !FalseQ[w]
&& FunctionOfQ[NonfreeFactors[Tan[w], x], (u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]] /; FreeQ[c, x] && LinearQ[v,
x] && IntegerQ[m + 1/2] &&  !SumQ[u] && InverseFunctionFreeQ[u, x]

Rule 898

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1262

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\csc ^2(x) \sec (x)}{\sqrt{\sin (2 x)} (-2+\tan (x))} \, dx &=\frac{\sin (x) \int \frac{\csc ^3(x) \sec (x) \sqrt{\tan (x)}}{-2+\tan (x)} \, dx}{\sqrt{\sin (2 x)} \sqrt{\tan (x)}}\\ &=\frac{\sin (x) \operatorname{Subst}\left (\int \frac{1+x^2}{(-2+x) x^{5/2}} \, dx,x,\tan (x)\right )}{\sqrt{\sin (2 x)} \sqrt{\tan (x)}}\\ &=\frac{(2 \sin (x)) \operatorname{Subst}\left (\int \frac{1+x^4}{x^4 \left (-2+x^2\right )} \, dx,x,\sqrt{\tan (x)}\right )}{\sqrt{\sin (2 x)} \sqrt{\tan (x)}}\\ &=\frac{(2 \sin (x)) \operatorname{Subst}\left (\int \left (-\frac{1}{2 x^4}-\frac{1}{4 x^2}+\frac{5}{4 \left (-2+x^2\right )}\right ) \, dx,x,\sqrt{\tan (x)}\right )}{\sqrt{\sin (2 x)} \sqrt{\tan (x)}}\\ &=\frac{\cos (x)}{2 \sqrt{\sin (2 x)}}+\frac{\cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}}+\frac{(5 \sin (x)) \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{\tan (x)}\right )}{2 \sqrt{\sin (2 x)} \sqrt{\tan (x)}}\\ &=\frac{\cos (x)}{2 \sqrt{\sin (2 x)}}+\frac{\cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{\tan (x)}}{\sqrt{2}}\right ) \sin (x)}{2 \sqrt{2} \sqrt{\sin (2 x)} \sqrt{\tan (x)}}\\ \end{align*}

Mathematica [C]  time = 5.90324, size = 119, normalized size = 1.72 \[ \frac{1}{4} \sqrt{\sin (2 x)} \left (5 \sqrt{\frac{\cos (x)}{2 \cos (x)-2}} \sqrt{\tan \left (\frac{x}{2}\right )} \sec (x) \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{2}\right )}}\right ),-1\right )+\Pi \left (-\frac{2}{-1+\sqrt{5}};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{2}\right )}}\right )\right |-1\right )+\Pi \left (\frac{1}{2} \left (-1+\sqrt{5}\right );\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{x}{2}\right )}}\right )\right |-1\right )\right )+\left (\frac{2 \cot (x)}{3}+1\right ) \csc (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x]^2*Sec[x])/(Sqrt[Sin[2*x]]*(-2 + Tan[x])),x]

[Out]

(Sqrt[Sin[2*x]]*((1 + (2*Cot[x])/3)*Csc[x] + 5*Sqrt[Cos[x]/(-2 + 2*Cos[x])]*(EllipticF[ArcSin[1/Sqrt[Tan[x/2]]
], -1] + EllipticPi[-2/(-1 + Sqrt[5]), -ArcSin[1/Sqrt[Tan[x/2]]], -1] + EllipticPi[(-1 + Sqrt[5])/2, -ArcSin[1
/Sqrt[Tan[x/2]]], -1])*Sec[x]*Sqrt[Tan[x/2]]))/4

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Maple [C]  time = 0.141, size = 396, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2*sec(x)/sin(2*x)^(1/2)/(-2+tan(x)),x)

[Out]

-1/480*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)/tan(1/2*x)^2*(-140*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(-2*tan(1/2
*x)+2)^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*(1+tan(1/2*x))^(1/2)*(-tan(1/2*x))^(1/2)*tan(1/2*x)+2
40*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*EllipticE((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*(1+
tan(1/2*x))^(1/2)*(-tan(1/2*x))^(1/2)*tan(1/2*x)+2^(1/2)*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^3-tan
(1/2*x))^(1/2)*sum((14*_alpha^3+3*_alpha^2+14*_alpha-11)*(_alpha^3+2*_alpha-3)*(1+tan(1/2*x))^(1/2)*(1-tan(1/2
*x))^(1/2)*(-tan(1/2*x))^(1/2)/(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*EllipticPi((1+tan(1/2*x))^(1/2),-1/4*_alpha
^3-1/2*_alpha+3/4,1/2*2^(1/2)),_alpha=RootOf(_Z^4+_Z^3+2*_Z^2-_Z+1))*tan(1/2*x)+40*(tan(1/2*x)*(tan(1/2*x)^2-1
))^(1/2)*tan(1/2*x)^4+120*tan(1/2*x)^3*(tan(1/2*x)^3-tan(1/2*x))^(1/2)-120*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*tan
(1/2*x)-40*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2))/(tan(1/2*x)^3-tan(1/2*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (x\right )^{2} \sec \left (x\right )}{{\left (\tan \left (x\right ) - 2\right )} \sqrt{\sin \left (2 \, x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2*sec(x)/sin(2*x)^(1/2)/(-2+tan(x)),x, algorithm="maxima")

[Out]

integrate(csc(x)^2*sec(x)/((tan(x) - 2)*sqrt(sin(2*x))), x)

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Fricas [B]  time = 2.64763, size = 431, normalized size = 6.25 \begin{align*} -\frac{4 \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (2 \, \cos \left (x\right ) + 3 \, \sin \left (x\right )\right )} - 4 \, \cos \left (x\right )^{2} - 15 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (4 \, \cos \left (x\right ) + 3 \, \sin \left (x\right )\right )} + \frac{1}{2} \, \cos \left (x\right )^{2} + \frac{7}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2}\right ) + 15 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right )^{2} + \frac{1}{2} \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} \sin \left (x\right ) - \frac{1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2}\right ) + 4}{48 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2*sec(x)/sin(2*x)^(1/2)/(-2+tan(x)),x, algorithm="fricas")

[Out]

-1/48*(4*sqrt(2)*sqrt(cos(x)*sin(x))*(2*cos(x) + 3*sin(x)) - 4*cos(x)^2 - 15*(cos(x)^2 - 1)*log(-1/2*sqrt(2)*s
qrt(cos(x)*sin(x))*(4*cos(x) + 3*sin(x)) + 1/2*cos(x)^2 + 7/2*cos(x)*sin(x) + 1/2) + 15*(cos(x)^2 - 1)*log(1/2
*cos(x)^2 + 1/2*sqrt(2)*sqrt(cos(x)*sin(x))*sin(x) - 1/2*cos(x)*sin(x) + 1/2) + 4)/(cos(x)^2 - 1)

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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(csc(x)**2*sec(x)/sin(2*x)**(1/2)/(-2+tan(x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (x\right )^{2} \sec \left (x\right )}{{\left (\tan \left (x\right ) - 2\right )} \sqrt{\sin \left (2 \, x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2*sec(x)/sin(2*x)^(1/2)/(-2+tan(x)),x, algorithm="giac")

[Out]

integrate(csc(x)^2*sec(x)/((tan(x) - 2)*sqrt(sin(2*x))), x)