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

Optimal. Leaf size=100 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{a^{5/2}}+\frac {43 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {11 \cot (x)}{16 a (a \csc (x)+a)^{3/2}}+\frac {\cot (x)}{4 (a \csc (x)+a)^{5/2}} \]

[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)

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Rubi [A]  time = 0.15, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3777, 3922, 3920, 3774, 203, 3795} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{a^{5/2}}+\frac {43 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {11 \cot (x)}{16 a (a \csc (x)+a)^{3/2}}+\frac {\cot (x)}{4 (a \csc (x)+a)^{5/2}} \]

Antiderivative was successfully verified.

[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 203

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

Rule 3774

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 3777

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 3795

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 3920

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 3922

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*} \int \frac {1}{(a+a \csc (x))^{5/2}} \, dx &=\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 \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^2}+\frac {43 \operatorname {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 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^{5/2}}+\frac {43 \tan ^{-1}\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*}

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Mathematica [A]  time = 0.51, size = 139, normalized size = 1.39 \[ \frac {\csc ^2(x) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (8 \sin (x)+15 \cos (2 x)-64 \sqrt {\csc (x)-1} \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4 \tan ^{-1}\left (\sqrt {\csc (x)-1}\right )+43 \sqrt {2} \sqrt {\csc (x)-1} \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4 \tan ^{-1}\left (\frac {\sqrt {\csc (x)-1}}{\sqrt {2}}\right )+7\right )}{32 (a (\csc (x)+1))^{5/2} \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )} \]

Antiderivative was successfully verified.

[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]))

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fricas [B]  time = 0.48, size = 546, normalized size = 5.46 \[ \left [-\frac {43 \, \sqrt {2} {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \sqrt {-a} \log \left (-\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} \sin \relax (x) - a \cos \relax (x)}{\sin \relax (x) + 1}\right ) + 32 \, {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \relax (x)^{2} + 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} + a \cos \relax (x) - {\left (2 \, a \cos \relax (x) + a\right )} \sin \relax (x) - a}{\cos \relax (x) + \sin \relax (x) + 1}\right ) - 2 \, {\left (15 \, \cos \relax (x)^{3} + 4 \, \cos \relax (x)^{2} - {\left (15 \, \cos \relax (x)^{2} + 11 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 15 \, \cos \relax (x) - 4\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}}{32 \, {\left (a^{3} \cos \relax (x)^{3} + 3 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3} + {\left (a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3}\right )} \sin \relax (x)\right )}}, \frac {43 \, \sqrt {2} {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right ) + 32 \, {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) - \sin \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right ) + {\left (15 \, \cos \relax (x)^{3} + 4 \, \cos \relax (x)^{2} - {\left (15 \, \cos \relax (x)^{2} + 11 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 15 \, \cos \relax (x) - 4\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}}{16 \, {\left (a^{3} \cos \relax (x)^{3} + 3 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3} + {\left (a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3}\right )} \sin \relax (x)\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[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))]

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giac [B]  time = 1.62, size = 348, normalized size = 3.48 \[ -\frac {1}{16} \, {\left (\frac {43 \, \sqrt {2} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, x\right )}}{\sqrt {a}}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} - \frac {16 \, {\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} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} - \frac {16 \, {\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} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} - \frac {8 \, {\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 )}{a^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} + \frac {8 \, {\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 )}{a^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} + \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 )}}{{\left (a \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{4} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(43*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/(a^(5/2)*sgn(tan(1/2*x) + 1)) - 16*(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*sgn(tan(1/2*x) +
1)) - 16*(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)))/sq
rt(abs(a)))/(a^4*sgn(tan(1/2*x) + 1)) - 8*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*ta
n(1/2*x))*sqrt(abs(a)) + abs(a))/(a^4*sgn(tan(1/2*x) + 1)) + 8*(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*sgn(tan(1/2*x) + 1)) + 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*sgn(tan(1/2*x) + 1)))*sgn(tan(1/2*x))

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maple [B]  time = 0.61, size = 1961, normalized size = 19.61 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(-1+cos(x))^2*(-11*2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-19*sin(x)*cos(x)^2*(-(-1+cos(x))/sin(x))^
(3/2)*2^(1/2)-11*sin(x)*cos(x)^2*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)+11*sin(x)*cos(x)^2*(-(-1+cos(x))/sin(x))^
(7/2)*2^(1/2)+19*sin(x)*cos(x)^2*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)-172*sin(x)*cos(x)^2*2^(1/2)*arctan((-(-1+
cos(x))/sin(x))^(1/2))-11*cos(x)^3*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)-11*cos(x)^2*(-(-1+cos(x))/sin(x))^(7/2)
*2^(1/2)+32*sin(x)*cos(x)^2*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+co
s(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))+128*sin(x)*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)
-128*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin
(x)-sin(x)+cos(x)-1))-512*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)-344*2^(1/2)*cos(x)*sin(x)*arctan((-(-1
+cos(x))/sin(x))^(1/2))-38*2^(1/2)*cos(x)*sin(x)*(-(-1+cos(x))/sin(x))^(3/2)-22*2^(1/2)*cos(x)*sin(x)*(-(-1+co
s(x))/sin(x))^(1/2)+38*sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+22*sin(x)*cos(x)*(-(-1+cos(x))/sin(x)
)^(7/2)*2^(1/2)-128*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/si
n(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))-512*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)+32*sin(x)*cos(x)^2*ln(-
(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x
)-cos(x)+1))+128*sin(x)*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)+11*sin(x)*(-(-1+cos(x))/sin(x))
^(7/2)*2^(1/2)-19*cos(x)^3*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+11*cos(x)*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)-1
9*cos(x)^2*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+19*sin(x)*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+19*cos(x)^3*(-(-1
+cos(x))/sin(x))^(3/2)*2^(1/2)+19*cos(x)*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+11*cos(x)^3*(-(-1+cos(x))/sin(x))
^(1/2)*2^(1/2)-19*cos(x)*(-(-1+cos(x))/sin(x))^(3/2)*2^(1/2)+172*cos(x)^3*2^(1/2)*arctan((-(-1+cos(x))/sin(x))
^(1/2))-11*cos(x)*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)+19*2^(1/2)*cos(x)^2*(-(-1+cos(x))/sin(x))^(3/2)+64*cos(x
)*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)
*sin(x)+sin(x)-cos(x)+1))+64*cos(x)*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2
^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))+256*cos(x)*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin
(x))^(1/2)+1)+256*cos(x)*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-344*2^(1/2)*cos(x)*arctan((-(-1+
cos(x))/sin(x))^(1/2))+688*2^(1/2)*sin(x)*arctan((-(-1+cos(x))/sin(x))^(1/2))-516*2^(1/2)*cos(x)^2*arctan((-(-
1+cos(x))/sin(x))^(1/2))-19*2^(1/2)*sin(x)*(-(-1+cos(x))/sin(x))^(3/2)+11*2^(1/2)*cos(x)^2*(-(-1+cos(x))/sin(x
))^(1/2)+688*2^(1/2)*arctan((-(-1+cos(x))/sin(x))^(1/2))+96*cos(x)^2*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*
sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))+96*cos(x)^2*ln(-(2^(1/2)
*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)
-1))+384*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)+384*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(
x))^(1/2)-1)+64*cos(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))
/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))+64*cos(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x
)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))+256*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin
(x))^(1/2)+1)+256*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-128*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/s
in(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))-128*sin(x)*
ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-s
in(x)+cos(x)-1))-512*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)-512*sin(x)*arctan(2^(1/2)*(-(-1+cos(
x))/sin(x))^(1/2)-1)-19*2^(1/2)*(-(-1+cos(x))/sin(x))^(3/2)-32*cos(x)^3*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/
2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))-128*cos(x)^3*arctan(2
^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-32*cos(x)^3*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x
)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))-128*cos(x)^3*arctan(2^(1/2)*(-(-1+cos(x))/s
in(x))^(1/2)+1)+19*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+11*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)-11*2^(1/2)*(-(-1
+cos(x))/sin(x))^(1/2))/(a*(1+sin(x))/sin(x))^(5/2)/sin(x)^5/(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \csc \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*csc(x) + a)^(-5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+\frac {a}{\sin \relax (x)}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a/sin(x))^(5/2),x)

[Out]

int(1/(a + a/sin(x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc {\relax (x )} + a\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csc(x) + a)**(-5/2), x)

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