3.84 \(\int \csc ^{-1}(x) \, dx\)

Optimal. Leaf size=17 \[ \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )+x \csc ^{-1}(x) \]

[Out]

x*arccsc(x)+arctanh((1-1/x^2)^(1/2))

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.000, Rules used = {5215, 266, 63, 206} \[ \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )+x \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[x],x]

[Out]

x*ArcCsc[x] + ArcTanh[Sqrt[1 - x^(-2)]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5215

Int[ArcCsc[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcCsc[c*x], x] + Dist[1/c, Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x], x
] /; FreeQ[c, x]

Rubi steps

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

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Mathematica [B]  time = 0.05, size = 64, normalized size = 3.76 \[ \frac {\sqrt {x^2-1} \left (\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )\right )}{2 \sqrt {1-\frac {1}{x^2}} x}+x \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[x],x]

[Out]

x*ArcCsc[x] + (Sqrt[-1 + x^2]*(-Log[1 - x/Sqrt[-1 + x^2]] + Log[1 + x/Sqrt[-1 + x^2]]))/(2*Sqrt[1 - x^(-2)]*x)

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fricas [B]  time = 0.44, size = 35, normalized size = 2.06 \[ {\left (x - 2\right )} \operatorname {arccsc}\relax (x) - 4 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) - \log \left (-x + \sqrt {x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x),x, algorithm="fricas")

[Out]

(x - 2)*arccsc(x) - 4*arctan(-x + sqrt(x^2 - 1)) - log(-x + sqrt(x^2 - 1))

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giac [B]  time = 0.03, size = 37, normalized size = 2.18 \[ x \arcsin \left (\frac {1}{x}\right ) + \frac {1}{2} \, \log \left (\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x),x, algorithm="giac")

[Out]

x*arcsin(1/x) + 1/2*log(sqrt(-1/x^2 + 1) + 1) - 1/2*log(-sqrt(-1/x^2 + 1) + 1)

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maple [A]  time = 0.01, size = 20, normalized size = 1.18 \[ x \,\mathrm {arccsc}\relax (x )+\ln \left (x +\sqrt {-\frac {1}{x^{2}}+1}\, x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(x),x)

[Out]

x*arccsc(x)+ln(x+(-1/x^2+1)^(1/2)*x)

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maxima [B]  time = 0.60, size = 35, normalized size = 2.06 \[ x \operatorname {arccsc}\relax (x) + \frac {1}{2} \, \log \left (\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {-\frac {1}{x^{2}} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(x),x, algorithm="maxima")

[Out]

x*arccsc(x) + 1/2*log(sqrt(-1/x^2 + 1) + 1) - 1/2*log(-sqrt(-1/x^2 + 1) + 1)

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mupad [B]  time = 0.21, size = 20, normalized size = 1.18 \[ x\,\mathrm {asin}\left (\frac {1}{x}\right )+\ln \left (x+\sqrt {x^2-1}\right )\,\mathrm {sign}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(1/x),x)

[Out]

x*asin(1/x) + log(x + (x^2 - 1)^(1/2))*sign(x)

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sympy [A]  time = 2.24, size = 17, normalized size = 1.00 \[ x \operatorname {acsc}{\relax (x )} + \begin {cases} \operatorname {acosh}{\relax (x )} & \text {for}\: \left |{x^{2}}\right | > 1 \\- i \operatorname {asin}{\relax (x )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(x),x)

[Out]

x*acsc(x) + Piecewise((acosh(x), Abs(x**2) > 1), (-I*asin(x), True))

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