Integrand size = 10, antiderivative size = 81 \[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^{3/2}}+\frac {5 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\cot (x)}{2 (a+a \csc (x))^{3/2}} \]
-2*arctan(cot(x)*a^(1/2)/(a+a*csc(x))^(1/2))/a^(3/2)+1/2*cot(x)/(a+a*csc(x ))^(3/2)+5/4*arctan(1/2*cot(x)*a^(1/2)*2^(1/2)/(a+a*csc(x))^(1/2))/a^(3/2) *2^(1/2)
Time = 0.50 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=-\frac {\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \left (2-2 \csc (x)+8 \arctan \left (\sqrt {-1+\csc (x)}\right ) \sqrt {-1+\csc (x)} (1+\csc (x))-5 \sqrt {2} \arctan \left (\frac {\sqrt {-1+\csc (x)}}{\sqrt {2}}\right ) \sqrt {-1+\csc (x)} \csc (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2\right )}{4 a (-1+\csc (x)) \sqrt {a (1+\csc (x))} \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )} \]
-1/4*((Cos[x/2] - Sin[x/2])*(2 - 2*Csc[x] + 8*ArcTan[Sqrt[-1 + Csc[x]]]*Sq rt[-1 + Csc[x]]*(1 + Csc[x]) - 5*Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]] *Sqrt[-1 + Csc[x]]*Csc[x]*(Cos[x/2] + Sin[x/2])^2))/(a*(-1 + Csc[x])*Sqrt[ a*(1 + Csc[x])]*(Cos[x/2] + Sin[x/2]))
Time = 0.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4264, 27, 3042, 4408, 3042, 4261, 216, 4282, 216}
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.
| \(\displaystyle \int \frac {1}{(a \csc (x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \csc (x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 4264 |
| \(\displaystyle \frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}-\frac {\int -\frac {4 a-a \csc (x)}{2 \sqrt {\csc (x) a+a}}dx}{2 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 a-a \csc (x)}{\sqrt {\csc (x) a+a}}dx}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 a-a \csc (x)}{\sqrt {\csc (x) a+a}}dx}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {4 \int \sqrt {\csc (x) a+a}dx-5 a \int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \int \sqrt {\csc (x) a+a}dx-5 a \int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {-8 a \int \frac {1}{\frac {a^2 \cot ^2(x)}{\csc (x) a+a}+a}d\frac {a \cot (x)}{\sqrt {\csc (x) a+a}}-5 a \int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-5 a \int \frac {\csc (x)}{\sqrt {\csc (x) a+a}}dx-8 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {10 a \int \frac {1}{\frac {a^2 \cot ^2(x)}{\csc (x) a+a}+2 a}d\frac {a \cot (x)}{\sqrt {\csc (x) a+a}}-8 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {5 \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )-8 \sqrt {a} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{4 a^2}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}}\) |
(-8*Sqrt[a]*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] + 5*Sqrt[2]*Sqrt[a ]*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/(4*a^2) + Cot[x]/ (2*(a + a*Csc[x])^(3/2))
3.1.17.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-Cot[c + d*x])*((a + b*Csc[c + d*x])^n/(d*(2*n + 1))), x] + Simp[1/(a^2*(2*n + 1)) Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && Int egerQ[2*n]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(60)=120\).
Time = 0.54 (sec) , antiderivative size = 642, normalized size of antiderivative = 7.93
1/4/(a/(1-cos(x))*(csc(x)*(1-cos(x))^2+2-2*cos(x)+sin(x)))^(3/2)*(csc(x)-c ot(x)+1)*(csc(x)^2*2^(1/2)*ln(-(csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2 )+1)/((csc(x)-cot(x))^(1/2)*2^(1/2)-csc(x)+cot(x)-1))*(1-cos(x))^2+4*csc(x )^2*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)*(1-cos(x))^2+4*csc(x)^ 2*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)*(1-cos(x))^2+csc(x)^2*2^ (1/2)*ln(-((csc(x)-cot(x))^(1/2)*2^(1/2)-csc(x)+cot(x)-1)/(csc(x)-cot(x)+( csc(x)-cot(x))^(1/2)*2^(1/2)+1))*(1-cos(x))^2+2*2^(1/2)*ln(-(csc(x)-cot(x) +(csc(x)-cot(x))^(1/2)*2^(1/2)+1)/((csc(x)-cot(x))^(1/2)*2^(1/2)-csc(x)+co t(x)-1))*(csc(x)-cot(x))+8*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1) *(csc(x)-cot(x))+8*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)*(csc(x) -cot(x))+2*2^(1/2)*ln(-((csc(x)-cot(x))^(1/2)*2^(1/2)-csc(x)+cot(x)-1)/(cs c(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1))*(csc(x)-cot(x))-10*csc(x)^2* arctan((csc(x)-cot(x))^(1/2))*(1-cos(x))^2-2*(csc(x)-cot(x))^(3/2)+2^(1/2) *ln(-(csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1)/((csc(x)-cot(x))^(1/2 )*2^(1/2)-csc(x)+cot(x)-1))+4*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2) +1)+4*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)+2^(1/2)*ln(-((csc(x) -cot(x))^(1/2)*2^(1/2)-csc(x)+cot(x)-1)/(csc(x)-cot(x)+(csc(x)-cot(x))^(1/ 2)*2^(1/2)+1))-20*arctan((csc(x)-cot(x))^(1/2))*(csc(x)-cot(x))-10*arctan( (csc(x)-cot(x))^(1/2))+2*(csc(x)-cot(x))^(1/2))/(csc(x)-cot(x))^(3/2)*2^(1 /2)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (60) = 120\).
Time = 0.28 (sec) , antiderivative size = 427, normalized size of antiderivative = 5.27 \[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=\left [-\frac {5 \, \sqrt {2} {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \sqrt {-a} \log \left (-\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} \sin \left (x\right ) - a \cos \left (x\right )}{\sin \left (x\right ) + 1}\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (x\right )^{2} + 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} + a \cos \left (x\right ) - {\left (2 \, a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) + 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{4 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}}, \frac {5 \, \sqrt {2} {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right ) - {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{2 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}}\right ] \]
[-1/4*(5*sqrt(2)*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(-a)*lo g(-(sqrt(2)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x))*sin(x) - a*cos(x))/(sin(x ) + 1)) + 4*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(-a)*log((2* a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x ) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt((a*sin(x) + a)/sin(x)) )/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x)), 1/2*( 5*sqrt(2)*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(a)*arctan(sqr t(2)*sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) + 1)/(a*cos(x) + a*sin(x) + a)) + 4*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(a)*arctan(-s qrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin (x) + a)) - (cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt((a*sin(x) + a)/sin(x )))/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))]
\[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=\int \frac {1}{\left (a \csc {\left (x \right )} + a\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (60) = 120\).
Time = 0.33 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=-\frac {\sqrt {2} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {3}{2}} - \sqrt {2} \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}}{2 \, {\left (a^{\frac {3}{2}} + \frac {2 \, a^{\frac {3}{2}} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a^{\frac {3}{2}} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}} + \frac {\sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right )\right )}}{a^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} \arctan \left (\sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}{2 \, a^{\frac {3}{2}}} \]
-1/2*(sqrt(2)*(sin(x)/(cos(x) + 1))^(3/2) - sqrt(2)*sqrt(sin(x)/(cos(x) + 1)))/(a^(3/2) + 2*a^(3/2)*sin(x)/(cos(x) + 1) + a^(3/2)*sin(x)^2/(cos(x) + 1)^2) + sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(sin(x)/(cos (x) + 1)))) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(sin(x)/(cos(x) + 1)))))/a^(3/2) - 5/2*sqrt(2)*arctan(sqrt(sin(x)/(cos(x) + 1)))/a^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (60) = 120\).
Time = 0.36 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.00 \[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=-\frac {5 \, \sqrt {2} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, x\right )}}{\sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} + \frac {{\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a^{3}} + \frac {{\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a^{3}} + \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{2 \, a^{3}} - \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{2 \, a^{3}} - \frac {\sqrt {2} {\left (\sqrt {a \tan \left (\frac {1}{2} \, x\right )} a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a\right )}}{2 \, {\left (a \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2} a} \]
-5/2*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/a^(3/2) + (a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/ 2*x)))/sqrt(abs(a)))/a^3 + (a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/2*sqr t(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/a^3 + 1/2 *(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2 *x))*sqrt(abs(a)) + abs(a))/a^3 - 1/2*(a*sqrt(abs(a)) - abs(a)^(3/2))*log( a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/a^3 - 1/2 *sqrt(2)*(sqrt(a*tan(1/2*x))*a*tan(1/2*x) - sqrt(a*tan(1/2*x))*a)/((a*tan( 1/2*x) + a)^2*a)
Timed out. \[ \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\sin \left (x\right )}\right )}^{3/2}} \,d x \]