3.49 \(\int (a \csc ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=46 \[ -\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \]

[Out]

-1/2*a^(3/2)*arctanh(cot(x)*a^(1/2)/(a*csc(x)^2)^(1/2))-1/2*a*cot(x)*(a*csc(x)^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 206} \[ -\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^2)^(3/2),x]

[Out]

-(a^(3/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]])/2 - (a*Cot[x]*Sqrt[a*Csc[x]^2])/2

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 206

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps

\begin {align*} \int \left (a \csc ^2(x)\right )^{3/2} \, dx &=-\left (a \operatorname {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right )\\ &=-\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 39, normalized size = 0.85 \[ -\frac {1}{2} a \sin (x) \sqrt {a \csc ^2(x)} \left (-\log \left (\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )\right )+\cot (x) \csc (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^2)^(3/2),x]

[Out]

-1/2*(a*Sqrt[a*Csc[x]^2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x])

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fricas [A]  time = 0.59, size = 49, normalized size = 1.07 \[ -\frac {{\left (2 \, a \cos \relax (x) + {\left (a \cos \relax (x)^{2} - a\right )} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}}}{4 \, \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^2)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(2*a*cos(x) + (a*cos(x)^2 - a)*log(-(cos(x) - 1)/(cos(x) + 1)))*sqrt(-a/(cos(x)^2 - 1))/sin(x)

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giac [B]  time = 0.28, size = 72, normalized size = 1.57 \[ \frac {1}{8} \, {\left (2 \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) - \frac {{\left (\frac {2 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} - \mathrm {sgn}\left (\sin \relax (x)\right )\right )} {\left (\cos \relax (x) + 1\right )}}{\cos \relax (x) - 1} - \frac {{\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1}\right )} a^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/8*(2*log(-(cos(x) - 1)/(cos(x) + 1))*sgn(sin(x)) - (2*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - sgn(sin(x)))*(
cos(x) + 1)/(cos(x) - 1) - (cos(x) - 1)*sgn(sin(x))/(cos(x) + 1))*a^(3/2)

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maple [A]  time = 0.46, size = 53, normalized size = 1.15 \[ -\frac {\left (\left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )+\cos \relax (x )\right ) \sin \relax (x ) \left (-\frac {a}{-1+\cos ^{2}\relax (x )}\right )^{\frac {3}{2}} \sqrt {4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csc(x)^2)^(3/2),x)

[Out]

-1/4*(cos(x)^2*ln(-(-1+cos(x))/sin(x))-ln(-(-1+cos(x))/sin(x))+cos(x))*sin(x)*(-1/(-1+cos(x)^2)*a)^(3/2)*4^(1/
2)

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maxima [B]  time = 0.54, size = 318, normalized size = 6.91 \[ -\frac {{\left (4 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \cos \relax (x) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \relax (x) - {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right ) + 2 \, {\left (a \sin \left (3 \, x\right ) + a \sin \relax (x)\right )} \cos \left (4 \, x\right ) - 2 \, {\left (a \cos \left (3 \, x\right ) + a \cos \relax (x)\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \sin \left (3 \, x\right ) + 2 \, a \sin \relax (x)\right )} \sqrt {-a}}{2 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(4*a*cos(3*x)*sin(2*x) + 4*a*cos(x)*sin(2*x) - 4*a*cos(2*x)*sin(x) - (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*s
in(4*x)^2 - 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2 - 2*(2*a*cos(2*x) - a)*cos(4*x) - 4*a*cos(2*x) + a)*arctan2
(sin(x), cos(x) + 1) + (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*sin(4*x)^2 - 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2
- 2*(2*a*cos(2*x) - a)*cos(4*x) - 4*a*cos(2*x) + a)*arctan2(sin(x), cos(x) - 1) + 2*(a*sin(3*x) + a*sin(x))*co
s(4*x) - 2*(a*cos(3*x) + a*cos(x))*sin(4*x) - 2*(2*a*cos(2*x) - a)*sin(3*x) + 2*a*sin(x))*sqrt(-a)/(2*(2*cos(2
*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) -
1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {a}{{\sin \relax (x)}^2}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/sin(x)^2)^(3/2),x)

[Out]

int((a/sin(x)^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csc(x)**2)**(3/2),x)

[Out]

Integral((a*csc(x)**2)**(3/2), x)

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