3.16 \(\int (1+\csc ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{1}{2} \cot (x) \sqrt{\cot ^2(x)+2}-\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right )-2 \sinh ^{-1}\left (\frac{\cot (x)}{\sqrt{2}}\right ) \]

[Out]

-2*ArcSinh[Cot[x]/Sqrt[2]] - ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]] - (Cot[x]*Sqrt[2 + Cot[x]^2])/2

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Rubi [A]  time = 0.0394003, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4128, 416, 523, 215, 377, 203} \[ -\frac{1}{2} \cot (x) \sqrt{\cot ^2(x)+2}-\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right )-2 \sinh ^{-1}\left (\frac{\cot (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Csc[x]^2)^(3/2),x]

[Out]

-2*ArcSinh[Cot[x]/Sqrt[2]] - ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]] - (Cot[x]*Sqrt[2 + Cot[x]^2])/2

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 215

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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])

Rubi steps

\begin{align*} \int \left (1+\csc ^2(x)\right )^{3/2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (2+x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{2} \cot (x) \sqrt{2+\cot ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{6+4 x^2}{\left (1+x^2\right ) \sqrt{2+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{2} \cot (x) \sqrt{2+\cot ^2(x)}-2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^2}} \, dx,x,\cot (x)\right )-\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{2+x^2}} \, dx,x,\cot (x)\right )\\ &=-2 \sinh ^{-1}\left (\frac{\cot (x)}{\sqrt{2}}\right )-\frac{1}{2} \cot (x) \sqrt{2+\cot ^2(x)}-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\cot (x)}{\sqrt{2+\cot ^2(x)}}\right )\\ &=-2 \sinh ^{-1}\left (\frac{\cot (x)}{\sqrt{2}}\right )-\tan ^{-1}\left (\frac{\cot (x)}{\sqrt{2+\cot ^2(x)}}\right )-\frac{1}{2} \cot (x) \sqrt{2+\cot ^2(x)}\\ \end{align*}

Mathematica [B]  time = 0.166718, size = 94, normalized size = 2.14 \[ \frac{\sin ^3(x) \left (\csc ^2(x)+1\right )^{3/2} \left (-2 \sqrt{2} \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right )-4 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right )+\sqrt{\cos (2 x)-3} \cot (x) \csc (x)\right )}{(\cos (2 x)-3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Csc[x]^2)^(3/2),x]

[Out]

((1 + Csc[x]^2)^(3/2)*(-4*Sqrt[2]*ArcTan[(Sqrt[2]*Cos[x])/Sqrt[-3 + Cos[2*x]]] + Sqrt[-3 + Cos[2*x]]*Cot[x]*Cs
c[x] - 2*Sqrt[2]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])*Sin[x]^3)/(-3 + Cos[2*x])^(3/2)

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Maple [B]  time = 0.261, size = 312, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+csc(x)^2)^(3/2),x)

[Out]

-1/2*((cos(x)^2-2)/(cos(x)^2-1))^(3/2)*(-1+cos(x))^2*(cos(x)*(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)+2*cos(x)*ln(-2
*(cos(x)^2*(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)+cos(x)^2+cos(x)-(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)-2)/sin(x)^2)+
2*cos(x)*arctan(cos(x)*(-1+cos(x))/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)/sin(x)^2)-2*cos(x)*arctanh((cos(x)^2-3*c
os(x)+2)/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)/sin(x)^2)-2*ln(-2*(cos(x)^2*(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)+cos
(x)^2+cos(x)-(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)-2)/sin(x)^2)-2*arctan(cos(x)*(-1+cos(x))/(-(cos(x)^2-2)/(cos(x
)+1)^2)^(1/2)/sin(x)^2)+2*arctanh((cos(x)^2-3*cos(x)+2)/(-(cos(x)^2-2)/(cos(x)+1)^2)^(1/2)/sin(x)^2))/sin(x)^3
/(-(cos(x)^2-2)/(cos(x)+1)^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\csc \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csc(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((csc(x)^2 + 1)^(3/2), x)

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Fricas [B]  time = 0.528504, size = 599, normalized size = 13.61 \begin{align*} \frac{\arctan \left (\frac{{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right ) - \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 1}\right ) \sin \left (x\right ) - \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \sin \left (x\right ) - 2 \, \log \left (-\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) -{\left (\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) - 1\right )} \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} + 2\right ) \sin \left (x\right ) + 2 \, \log \left (-\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) -{\left (\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) - 1\right )} \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} + 2\right ) \sin \left (x\right ) - \sqrt{\frac{\cos \left (x\right )^{2} - 2}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right )}{2 \, \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csc(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*(arctan(((cos(x)^3 - cos(x))*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*sin(x) - cos(x)*sin(x))/(cos(x)^4 - 3*cos
(x)^2 + 1))*sin(x) - arctan(sin(x)/cos(x))*sin(x) - 2*log(-cos(x)^2 + cos(x)*sin(x) - (cos(x)^2 - cos(x)*sin(x
) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2)*sin(x) + 2*log(-cos(x)^2 - cos(x)*sin(x) - (cos(x)^2 + cos(x)*
sin(x) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2)*sin(x) - sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*cos(x))/sin(
x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\csc ^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csc(x)**2)**(3/2),x)

[Out]

Integral((csc(x)**2 + 1)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\csc \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csc(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((csc(x)^2 + 1)^(3/2), x)