\(\int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx\) [634]

Optimal result

Integrand size = 21, antiderivative size = 69 \[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=\frac {\cos (x)}{2 \sqrt {\sin (2 x)}}+\frac {\cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right ) \sin (x)}{2 \sqrt {2} \sqrt {\sin (2 x)} \sqrt {\tan (x)}} \]

[Out]

1/2*cos(x)/sin(2*x)^(1/2)+1/3*cos(x)*cot(x)/sin(2*x)^(1/2)-5/4*arctanh(1/2*tan(x)^(1/2)*2^(1/2))*sin(x)*2^(1/2
)/sin(2*x)^(1/2)/tan(x)^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4475, 912, 1276, 213} \[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=-\frac {5 \sin (x) \text {arctanh}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right )}{2 \sqrt {2} \sqrt {\sin (2 x)} \sqrt {\tan (x)}}+\frac {\cos (x)}{2 \sqrt {\sin (2 x)}}+\frac {\cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}} \]

[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 213

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

Rule 912

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 1276

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 4475

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]

Rubi steps \begin{align*} \text {integral}& = \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) \text {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)) \text {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)) \text {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)) \text {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 \text {arctanh}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right ) \sin (x)}{2 \sqrt {2} \sqrt {\sin (2 x)} \sqrt {\tan (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16 \[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=\frac {1}{4} \sqrt {\sin (2 x)} \left (\left (1+\frac {2 \cot (x)}{3}\right ) \csc (x)-\frac {5 \arctan \left (\frac {\sqrt {\tan \left (\frac {x}{2}\right )}}{\sqrt {-1+\tan ^2\left (\frac {x}{2}\right )}}\right ) \sqrt {-\frac {\cos (x)}{2+2 \cos (x)}} \sec (x)}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right ) \]

[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*ArcTan[Sqrt[Tan[x/2]]/Sqrt[-1 + Tan[x/2]^2]]*Sqrt[-(Cos[x]/(2
+ 2*Cos[x]))]*Sec[x])/Sqrt[Tan[x/2]]))/4

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.62 (sec) , antiderivative size = 396, normalized size of antiderivative = 5.74

method	result	size
default	\(-\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2}-1}}\, \left (-140 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \operatorname {EllipticF}\left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )+240 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \operatorname {EllipticE}\left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )+\sqrt {2}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+2 \textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (14 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}+14 \underline {\hspace {1.25 ex}}\alpha -11\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha -3\right ) \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {1-\tan \left (\frac {x}{2}\right )}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticPi}\left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, -\frac {1}{4} \underline {\hspace {1.25 ex}}\alpha ^{3}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha +\frac {3}{4}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}}\right ) \tan \left (\frac {x}{2}\right )+40 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \tan \left (\frac {x}{2}\right )^{4}+120 \tan \left (\frac {x}{2}\right )^{3} \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}-120 \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )-40 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\right )}{480 \tan \left (\frac {x}{2}\right )^{2} \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}}\)	\(396\)

[In]

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

[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((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)+1)^(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((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(ta
n(1/2*x)+1)^(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)*(tan(1/2*x)+1)^(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((tan(1/2*x)+1)^(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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (50) = 100\).

Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.74 \[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=-\frac {4 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (2 \, \cos \left (x\right ) + 3 \, \sin \left (x\right )\right )} - 4 \, \cos \left (x\right )^{2} - 15 \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (4 \, \cos \left (x\right ) + 3 \, \sin \left (x\right )\right )} + \frac {1}{2} \, \cos \left (x\right )^{2} + \frac {7}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right )^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} \sin \left (x\right ) - \frac {1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{2}\right ) + 4}{48 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]

[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)

Sympy [F]

\[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=\int \frac {\csc ^{2}{\left (x \right )} \sec {\left (x \right )}}{\left (\tan {\left (x \right )} - 2\right ) \sqrt {\sin {\left (2 x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=\int { \frac {\csc \left (x\right )^{2} \sec \left (x\right )}{{\left (\tan \left (x\right ) - 2\right )} \sqrt {\sin \left (2 \, x\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(x) \sec (x)}{\sqrt {\sin (2 x)} (-2+\tan (x))} \, dx=\int \frac {1}{\sqrt {\sin \left (2\,x\right )}\,\cos \left (x\right )\,{\sin \left (x\right )}^2\,\left (\mathrm {tan}\left (x\right )-2\right )} \,d x \]

[In]

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

[Out]

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