3.25 \(\int (-1-\csc ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} \cot (x) \sqrt{-\cot ^2(x)-2}-2 \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right )-\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.045857, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4128, 416, 523, 217, 203, 377, 206} \[ \frac{1}{2} \cot (x) \sqrt{-\cot ^2(x)-2}-2 \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right )-\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 - Csc[x]^2)^(3/2),x]

[Out]

-2*ArcTan[Cot[x]/Sqrt[-2 - Cot[x]^2]] - ArcTanh[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 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 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 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 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])

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

Mathematica [A]  time = 0.112136, size = 96, normalized size = 1.75 \[ \frac{\sin ^3(x) \left (-\csc ^2(x)-1\right )^{3/2} \left (-2 \sqrt{2} \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right )-4 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right )+\sqrt{\cos (2 x)-3} \cot (x) \csc (x)\right )}{(\cos (2 x)-3)^{3/2}} \]

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]*C
sc[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 = 0.254, size = 268, normalized size = 4.9 \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]

1/4*(-(cos(x)^2-2)/(cos(x)^2-1))^(3/2)*4^(1/2)*(-1+cos(x))^2*(2*cos(x)*arctan(1/4*(-1+cos(x))*(3*cos(x)*4^(1/2
)-2*cos(x)-3*4^(1/2)-2)/sin(x)^2/((cos(x)^2-2)/(cos(x)+1)^2)^(1/2))-2*cos(x)*arcsin(1/2*2^(1/2)*(cos(x)+2)/(co
s(x)+1))-2*cos(x)*arctanh(1/2*cos(x)*4^(1/2)*(-1+cos(x))/sin(x)^2/((cos(x)^2-2)/(cos(x)+1)^2)^(1/2))+cos(x)*((
cos(x)^2-2)/(cos(x)+1)^2)^(1/2)-2*arctan(1/4*(-1+cos(x))*(3*cos(x)*4^(1/2)-2*cos(x)-3*4^(1/2)-2)/sin(x)^2/((co
s(x)^2-2)/(cos(x)+1)^2)^(1/2))+2*arcsin(1/2*2^(1/2)*(cos(x)+2)/(cos(x)+1))+2*arctanh(1/2*cos(x)*4^(1/2)*(-1+co
s(x))/sin(x)^2/((cos(x)^2-2)/(cos(x)+1)^2)^(1/2)))/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 [C]  time = 0.500099, size = 714, normalized size = 12.98 \begin{align*} \frac{{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )} \log \left (-2 \, \sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1}{\left (e^{\left (2 i \, x\right )} - 1\right )} + 2 \, e^{\left (4 i \, x\right )} - 8 \, e^{\left (2 i \, x\right )} - 2\right ) +{\left (4 i \, e^{\left (4 i \, x\right )} - 8 i \, e^{\left (2 i \, x\right )} + 4 i\right )} \log \left (\sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} + 2 i + 1\right ) -{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )} \log \left (\sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} + 1\right ) +{\left (-4 i \, e^{\left (4 i \, x\right )} + 8 i \, e^{\left (2 i \, x\right )} - 4 i\right )} \log \left (\sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1} - e^{\left (2 i \, x\right )} - 2 i + 1\right ) - \sqrt{e^{\left (4 i \, x\right )} - 6 \, e^{\left (2 i \, x\right )} + 1}{\left (e^{\left (2 i \, x\right )} + 1\right )} - e^{\left (4 i \, x\right )} + 2 \, e^{\left (2 i \, x\right )} - 1}{2 \,{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\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*((e^(4*I*x) - 2*e^(2*I*x) + 1)*log(-2*sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1)*(e^(2*I*x) - 1) + 2*e^(4*I*x) - 8*
e^(2*I*x) - 2) + (4*I*e^(4*I*x) - 8*I*e^(2*I*x) + 4*I)*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 2*I
+ 1) - (e^(4*I*x) - 2*e^(2*I*x) + 1)*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 1) + (-4*I*e^(4*I*x)
+ 8*I*e^(2*I*x) - 4*I)*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) - 2*I + 1) - sqrt(e^(4*I*x) - 6*e^(2
*I*x) + 1)*(e^(2*I*x) + 1) - e^(4*I*x) + 2*e^(2*I*x) - 1)/(e^(4*I*x) - 2*e^(2*I*x) + 1)

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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)