Integrand size = 25, antiderivative size = 208 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\left (8 a^2+40 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2+40 a b+35 b^2}{24 a^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\csc ^4(e+f x)}{4 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sin ^2(e+f x)}} \]
-1/8*(8*a^2+40*a*b+35*b^2)*arctanh((a+b*sin(f*x+e)^2)^(1/2)/a^(1/2))/a^(9/ 2)/f+1/24*(8*a^2+40*a*b+35*b^2)/a^3/f/(a+b*sin(f*x+e)^2)^(3/2)+1/8*(8*a+7* b)*csc(f*x+e)^2/a^2/f/(a+b*sin(f*x+e)^2)^(3/2)-1/4*csc(f*x+e)^4/a/f/(a+b*s in(f*x+e)^2)^(3/2)+1/8*(8*a^2+40*a*b+35*b^2)/a^4/f/(a+b*sin(f*x+e)^2)^(1/2 )
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.88 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.56 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {3 a \csc ^4(e+f x) \left (8 a+7 b-2 a \csc ^2(e+f x)\right )+\left (8 a^2+40 a b+35 b^2\right ) \csc ^2(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \sin ^2(e+f x)}{a}\right )}{24 a^3 f \left (b+a \csc ^2(e+f x)\right ) \sqrt {a+b \sin ^2(e+f x)}} \]
(3*a*Csc[e + f*x]^4*(8*a + 7*b - 2*a*Csc[e + f*x]^2) + (8*a^2 + 40*a*b + 3 5*b^2)*Csc[e + f*x]^2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sin[e + f*x] ^2)/a])/(24*a^3*f*(b + a*Csc[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])
Time = 0.35 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3673, 100, 27, 87, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^5 \left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3673 |
| \(\displaystyle \frac {\int \frac {\csc ^6(e+f x) \left (1-\sin ^2(e+f x)\right )^2}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^4(e+f x) \left (-4 a \sin ^2(e+f x)+8 a+7 b\right )}{2 \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin ^2(e+f x)}{2 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^4(e+f x) \left (-4 a \sin ^2(e+f x)+8 a+7 b\right )}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin ^2(e+f x)}{4 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+5 b (8 a+7 b)\right ) \int \frac {\csc ^2(e+f x)}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin ^2(e+f x)}{2 a}-\frac {(8 a+7 b) \csc ^2(e+f x)}{a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+5 b (8 a+7 b)\right ) \left (\frac {\int \frac {\csc ^2(e+f x)}{\left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin ^2(e+f x)}{a}+\frac {2}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{2 a}-\frac {(8 a+7 b) \csc ^2(e+f x)}{a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+5 b (8 a+7 b)\right ) \left (\frac {\frac {\int \frac {\csc ^2(e+f x)}{\sqrt {b \sin ^2(e+f x)+a}}d\sin ^2(e+f x)}{a}+\frac {2}{a \sqrt {a+b \sin ^2(e+f x)}}}{a}+\frac {2}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{2 a}-\frac {(8 a+7 b) \csc ^2(e+f x)}{a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+5 b (8 a+7 b)\right ) \left (\frac {\frac {2 \int \frac {1}{\frac {\sin ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \sin ^2(e+f x)+a}}{a b}+\frac {2}{a \sqrt {a+b \sin ^2(e+f x)}}}{a}+\frac {2}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{2 a}-\frac {(8 a+7 b) \csc ^2(e+f x)}{a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+5 b (8 a+7 b)\right ) \left (\frac {\frac {2}{a \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{2 a}-\frac {(8 a+7 b) \csc ^2(e+f x)}{a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\csc ^4(e+f x)}{2 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{2 f}\) |
(-1/2*Csc[e + f*x]^4/(a*(a + b*Sin[e + f*x]^2)^(3/2)) - (-(((8*a + 7*b)*Cs c[e + f*x]^2)/(a*(a + b*Sin[e + f*x]^2)^(3/2))) - ((8*a^2 + 5*b*(8*a + 7*b ))*(2/(3*a*(a + b*Sin[e + f*x]^2)^(3/2)) + ((-2*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/a^(3/2) + 2/(a*Sqrt[a + b*Sin[e + f*x]^2]))/a))/(2*a))/ (4*a))/(2*f)
3.6.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1)/2)/(2*f) Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m + 1 )/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && Integ erQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(184)=368\).
Time = 3.51 (sec) , antiderivative size = 988, normalized size of antiderivative = 4.75
(-1/4/a^3/sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2)+11/8/a^4*b/sin(f*x+e)^2*(a +b*sin(f*x+e)^2)^(1/2)-35/8/a^(9/2)*b^2*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^ 2)^(1/2))/sin(f*x+e))+1/a^3/sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)-5/a^(7/2 )*b*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin(f*x+e))-1/a^(5/2)*ln(( 2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin(f*x+e))-1/2*(a^2+4*a*b+3*b^2)/ a^4/(-a*b)^(1/2)/(sin(f*x+e)+(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b) ^(1/2)+1/2*(a^2+4*a*b+3*b^2)/a^4/(-a*b)^(1/2)/(sin(f*x+e)-(-a*b)^(1/2)/b)* (-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/a^2/b/(sin(f*x+e)-(-a*b)^(1/2)/b) ^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/6/a^3/(sin(f*x+e)-(-a*b)^(1/2)/b) ^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/a^4*b/(sin(f*x+e)-(-a*b)^(1/2) /b)^2*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)+1/12/a^2/(-a*b)^(1/2)/(sin(f*x+e )-(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)+1/6/a^3/(-a*b)^(1/2) *b/(sin(f*x+e)-(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)+1/12/a^ 4/(-a*b)^(1/2)/(sin(f*x+e)-(-a*b)^(1/2)/b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^( 1/2)*b^2-1/12/a^2/b/(sin(f*x+e)+(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b+b^ 2)/b)^(1/2)-1/6/a^3/(sin(f*x+e)+(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b+b^ 2)/b)^(1/2)-1/12/a^4*b/(sin(f*x+e)+(-a*b)^(1/2)/b)^2*(-b*cos(f*x+e)^2+(a*b +b^2)/b)^(1/2)-1/12/a^2/(-a*b)^(1/2)/(sin(f*x+e)+(-a*b)^(1/2)/b)*(-b*cos(f *x+e)^2+(a*b+b^2)/b)^(1/2)-1/6/a^3/(-a*b)^(1/2)*b/(sin(f*x+e)+(-a*b)^(1/2) /b)*(-b*cos(f*x+e)^2+(a*b+b^2)/b)^(1/2)-1/12/a^4/(-a*b)^(1/2)/(sin(f*x+...
Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (184) = 368\).
Time = 0.43 (sec) , antiderivative size = 984, normalized size of antiderivative = 4.73 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/48*(3*((8*a^2*b^2 + 40*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b + 56 *a^2*b^2 + 115*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (8*a^4 + 88*a^3*b + 323*a^ 2*b^2 + 450*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 8*a^4 + 56*a^3*b + 123*a^2*b ^2 + 110*a*b^3 + 35*b^4 - 2*(8*a^4 + 64*a^3*b + 171*a^2*b^2 + 185*a*b^3 + 70*b^4)*cos(f*x + e)^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f* x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(3*(8*a^3*b + 40*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^6 - (32*a^4 + 232*a^3*b + 500*a^2*b ^2 + 315*a*b^3)*cos(f*x + e)^4 - 50*a^4 - 205*a^3*b - 260*a^2*b^2 - 105*a* b^3 + (88*a^4 + 413*a^3*b + 640*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^2)*sqrt( -b*cos(f*x + e)^2 + a + b))/(a^5*b^2*f*cos(f*x + e)^8 - 2*(a^6*b + 2*a^5*b ^2)*f*cos(f*x + e)^6 + (a^7 + 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 - 2*(a ^7 + 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^2 + (a^7 + 2*a^6*b + a^5*b^2)*f), 1/24*(3*((8*a^2*b^2 + 40*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b + 56 *a^2*b^2 + 115*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (8*a^4 + 88*a^3*b + 323*a^ 2*b^2 + 450*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 8*a^4 + 56*a^3*b + 123*a^2*b ^2 + 110*a*b^3 + 35*b^4 - 2*(8*a^4 + 64*a^3*b + 171*a^2*b^2 + 185*a*b^3 + 70*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sq rt(-a)/a) - (3*(8*a^3*b + 40*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^6 - (32*a^4 + 232*a^3*b + 500*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^4 - 50*a^4 - 205*a^3*b - 260*a^2*b^2 - 105*a*b^3 + (88*a^4 + 413*a^3*b + 640*a^2*b^2 + 315*a*...
\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.25 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\frac {24 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} + \frac {120 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {7}{2}}} + \frac {105 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {9}{2}}} - \frac {24}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} - \frac {8}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a} - \frac {120 \, b}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{3}} - \frac {40 \, b}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {105 \, b^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{4}} - \frac {35 \, b^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {24}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \sin \left (f x + e\right )^{2}} - \frac {21 \, b}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2} \sin \left (f x + e\right )^{2}} + \frac {6}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \sin \left (f x + e\right )^{4}}}{24 \, f} \]
-1/24*(24*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e))))/a^(5/2) + 120*b*arcsinh (a/(sqrt(a*b)*abs(sin(f*x + e))))/a^(7/2) + 105*b^2*arcsinh(a/(sqrt(a*b)*a bs(sin(f*x + e))))/a^(9/2) - 24/(sqrt(b*sin(f*x + e)^2 + a)*a^2) - 8/((b*s in(f*x + e)^2 + a)^(3/2)*a) - 120*b/(sqrt(b*sin(f*x + e)^2 + a)*a^3) - 40* b/((b*sin(f*x + e)^2 + a)^(3/2)*a^2) - 105*b^2/(sqrt(b*sin(f*x + e)^2 + a) *a^4) - 35*b^2/((b*sin(f*x + e)^2 + a)^(3/2)*a^3) - 24/((b*sin(f*x + e)^2 + a)^(3/2)*a*sin(f*x + e)^2) - 21*b/((b*sin(f*x + e)^2 + a)^(3/2)*a^2*sin( f*x + e)^2) + 6/((b*sin(f*x + e)^2 + a)^(3/2)*a*sin(f*x + e)^4))/f
Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \]