\(\int \frac {1}{(a+a \csc (x))^{5/2}} \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 100 \[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^{5/2}}+\frac {43 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac {11 \cot (x)}{16 a (a+a \csc (x))^{3/2}} \]

[Out]

-2*arctan(cot(x)*a^(1/2)/(a+a*csc(x))^(1/2))/a^(5/2)+1/4*cot(x)/(a+a*csc(x))^(5/2)+11/16*cot(x)/a/(a+a*csc(x))
^(3/2)+43/32*arctan(1/2*cot(x)*a^(1/2)*2^(1/2)/(a+a*csc(x))^(1/2))/a^(5/2)*2^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3862, 4007, 4005, 3859, 209, 3880} \[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{a^{5/2}}+\frac {43 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {11 \cot (x)}{16 a (a \csc (x)+a)^{3/2}}+\frac {\cot (x)}{4 (a \csc (x)+a)^{5/2}} \]

[In]

Int[(a + a*Csc[x])^(-5/2),x]

[Out]

(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(5/2) + (43*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc
[x]])])/(16*Sqrt[2]*a^(5/2)) + Cot[x]/(4*(a + a*Csc[x])^(5/2)) + (11*Cot[x])/(16*a*(a + a*Csc[x])^(3/2))

Rule 209

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

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3862

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] + Dist[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] && IntegerQ[2*n]

Rule 3880

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-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
]

Rule 4005

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

Rule 4007

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(-(b
*c - a*d))*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1))), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[
e + f*x])^(m + 1)*Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f
}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = \frac {\cot (x)}{4 (a+a \csc (x))^{5/2}}-\frac {\int \frac {-4 a+\frac {3}{2} a \csc (x)}{(a+a \csc (x))^{3/2}} \, dx}{4 a^2} \\ & = \frac {\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac {11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}+\frac {\int \frac {8 a^2-\frac {11}{4} a^2 \csc (x)}{\sqrt {a+a \csc (x)}} \, dx}{8 a^4} \\ & = \frac {\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac {11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}+\frac {\int \sqrt {a+a \csc (x)} \, dx}{a^3}-\frac {43 \int \frac {\csc (x)}{\sqrt {a+a \csc (x)}} \, dx}{32 a^2} \\ & = \frac {\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac {11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^2}+\frac {43 \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{16 a^2} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^{5/2}}+\frac {43 \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac {11 \cot (x)}{16 a (a+a \csc (x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=\frac {\csc ^2(x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (7+15 \cos (2 x)-64 \arctan \left (\sqrt {-1+\csc (x)}\right ) \sqrt {-1+\csc (x)} \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+43 \sqrt {2} \arctan \left (\frac {\sqrt {-1+\csc (x)}}{\sqrt {2}}\right ) \sqrt {-1+\csc (x)} \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+8 \sin (x)\right )}{32 (a (1+\csc (x)))^{5/2} \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )} \]

[In]

Integrate[(a + a*Csc[x])^(-5/2),x]

[Out]

(Csc[x]^2*(Cos[x/2] + Sin[x/2])*(7 + 15*Cos[2*x] - 64*ArcTan[Sqrt[-1 + Csc[x]]]*Sqrt[-1 + Csc[x]]*(Cos[x/2] +
Sin[x/2])^4 + 43*Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]]*Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2])^4 + 8*Sin[x
]))/(32*(a*(1 + Csc[x]))^(5/2)*(Cos[x/2] - Sin[x/2]))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1103\) vs. \(2(75)=150\).

Time = 0.59 (sec) , antiderivative size = 1104, normalized size of antiderivative = 11.04

method	result	size
default	\(\text {Expression too large to display}\)	\(1104\)

[In]

int(1/(a+a*csc(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/16/(a/(1-cos(x))*(csc(x)*(1-cos(x))^2+2-2*cos(x)+sin(x)))^(5/2)*(csc(x)-cot(x)+1)*(16*2^(1/2)*ln(-(csc(x)-co
t(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))*(csc(x)-cot(x))+64*2^(1
/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)*(csc(x)-cot(x))+64*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)
*(csc(x)-cot(x))-258*csc(x)^2*arctan((csc(x)-cot(x))^(1/2))*(1-cos(x))^2+19*(csc(x)-cot(x))^(3/2)-11*(csc(x)-c
ot(x))^(7/2)-43*arctan((csc(x)-cot(x))^(1/2))-19*(csc(x)-cot(x))^(5/2)+16*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))*(csc(x)-cot(x))-172*csc(x)^3*arctan((
csc(x)-cot(x))^(1/2))*(1-cos(x))^3+24*csc(x)^2*2^(1/2)*ln(-(csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1)/((c
sc(x)-cot(x))^(1/2)*2^(1/2)-csc(x)+cot(x)-1))*(1-cos(x))^2+96*csc(x)^2*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^
(1/2)+1)*(1-cos(x))^2+96*csc(x)^2*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)*(1-cos(x))^2+24*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-c
os(x))^2-172*arctan((csc(x)-cot(x))^(1/2))*(csc(x)-cot(x))+64*csc(x)^3*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)
*2^(1/2)*(1-cos(x))^3+64*csc(x)^3*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)*2^(1/2)*(1-cos(x))^3+16*csc(x)^3*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))*2^(1/2)*(1-c
os(x))^3+4*csc(x)^4*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))*2^(1/2)*(1-cos(x))^4+16*csc(x)^4*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)*2^(1/2)*(1-cos(x))^4+16*cs
c(x)^4*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)*2^(1/2)*(1-cos(x))^4+4*csc(x)^4*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))*2^(1/2)*(1-cos(x))^4+16*csc(x)^3*ln(-(cs
c(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))*2^(1/2)*(1-cos(x
))^3-43*csc(x)^4*arctan((csc(x)-cot(x))^(1/2))*(1-cos(x))^4+4*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))+16*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1
)+4*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
))+16*2^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)+11*(csc(x)-cot(x))^(1/2))/(csc(x)-cot(x))^(5/2)*2^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (75) = 150\).

Time = 0.27 (sec) , antiderivative size = 546, normalized size of antiderivative = 5.46 \[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=\left [-\frac {43 \, \sqrt {2} {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\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 ) + 32 \, {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\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 (15 \, \cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - {\left (15 \, \cos \left (x\right )^{2} + 11 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 15 \, \cos \left (x\right ) - 4\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{32 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}}, \frac {43 \, \sqrt {2} {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\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 ) + 32 \, {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\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 (15 \, \cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - {\left (15 \, \cos \left (x\right )^{2} + 11 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 15 \, \cos \left (x\right ) - 4\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{16 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}}\right ] \]

[In]

integrate(1/(a+a*csc(x))^(5/2),x, algorithm="fricas")

[Out]

[-1/32*(43*sqrt(2)*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(-a)*log(-(sq
rt(2)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x))*sin(x) - a*cos(x))/(sin(x) + 1)) + 32*(cos(x)^3 + 3*cos(x)^2 + (cos
(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*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*(
15*cos(x)^3 + 4*cos(x)^2 - (15*cos(x)^2 + 11*cos(x) - 4)*sin(x) - 15*cos(x) - 4)*sqrt((a*sin(x) + a)/sin(x)))/
(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x)), 1/16*(4
3*sqrt(2)*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(a)*arctan(sqrt(2)*sqr
t(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) + 1)/(a*cos(x) + a*sin(x) + a)) + 32*(cos(x)^3 + 3*cos(x)^2 + (cos(x)
^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x)
+ 1)/(a*cos(x) + a*sin(x) + a)) + (15*cos(x)^3 + 4*cos(x)^2 - (15*cos(x)^2 + 11*cos(x) - 4)*sin(x) - 15*cos(x
) - 4)*sqrt((a*sin(x) + a)/sin(x)))/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*
a^3*cos(x) - 4*a^3)*sin(x))]

Sympy [F]

\[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=\int \frac {1}{\left (a \csc {\left (x \right )} + a\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(a+a*csc(x))**(5/2),x)

[Out]

Integral((a*csc(x) + a)**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \csc \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a+a*csc(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((a*csc(x) + a)^(-5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (75) = 150\).

Time = 0.36 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.86 \[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=-\frac {43 \, \sqrt {2} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, x\right )}}{\sqrt {a}}\right )}{16 \, a^{\frac {5}{2}}} + \frac {{\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a^{4}} + \frac {{\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a^{4}} + \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{2 \, a^{4}} - \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{2 \, a^{4}} - \frac {\sqrt {2} {\left (11 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 19 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 19 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{3} \tan \left (\frac {1}{2} \, x\right ) - 11 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{3}\right )}}{16 \, {\left (a \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{4} a^{2}} \]

[In]

integrate(1/(a+a*csc(x))^(5/2),x, algorithm="giac")

[Out]

-43/16*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/a^(5/2) + (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^4 + (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^4 + 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^4 - 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^4 - 1/16*sqrt(2)*(11*sqrt(a*tan(1/2*x
))*a^3*tan(1/2*x)^3 + 19*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x)^2 - 19*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x) - 11*sqrt(
a*tan(1/2*x))*a^3)/((a*tan(1/2*x) + a)^4*a^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\sin \left (x\right )}\right )}^{5/2}} \,d x \]

[In]

int(1/(a + a/sin(x))^(5/2),x)

[Out]

int(1/(a + a/sin(x))^(5/2), x)