\(\int x^{-1+n} \csc ^{-1}(a+b x^n) \, dx\) [39]

Optimal result

Integrand size = 14, antiderivative size = 48 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6847, 5359, 379, 272, 65, 212} \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n}+\frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n} \]

[In]

Int[x^(-1 + n)*ArcCsc[a + b*x^n],x]

[Out]

((a + b*x^n)*ArcCsc[a + b*x^n])/(b*n) + ArcTanh[Sqrt[1 - (a + b*x^n)^(-2)]]/(b*n)

Rule 65

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 212

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

Rule 272

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 379

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 5359

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

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \csc ^{-1}(a+b x) \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}} \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x} \, dx,x,a+b x^n\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{\left (a+b x^n\right )^2}\right )}{2 b n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(130\) vs. \(2(48)=96\).

Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.71 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\sqrt {-1+\left (a+b x^n\right )^2} \left (-\log \left (1-\frac {a+b x^n}{\sqrt {-1+\left (a+b x^n\right )^2}}\right )+\log \left (1+\frac {a+b x^n}{\sqrt {-1+\left (a+b x^n\right )^2}}\right )\right )}{2 b n \left (a+b x^n\right ) \sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}} \]

[In]

Integrate[x^(-1 + n)*ArcCsc[a + b*x^n],x]

[Out]

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

Maple [F]

\[\int x^{-1+n} \operatorname {arccsc}\left (a +b \,x^{n}\right )d x\]

[In]

int(x^(-1+n)*arccsc(a+b*x^n),x)

[Out]

int(x^(-1+n)*arccsc(a+b*x^n),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.96 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {b x^{n} \operatorname {arccsc}\left (b x^{n} + a\right ) - 2 \, a \arctan \left (-b x^{n} - a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right ) - \log \left (-b x^{n} - a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{b n} \]

[In]

integrate(x^(-1+n)*arccsc(a+b*x^n),x, algorithm="fricas")

[Out]

(b*x^n*arccsc(b*x^n + a) - 2*a*arctan(-b*x^n - a + sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2 - 1)) - log(-b*x^n - a +
sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2 - 1)))/(b*n)

Sympy [F(-1)]

Timed out. \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\text {Timed out} \]

[In]

integrate(x**(-1+n)*acsc(a+b*x**n),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arccsc}\left (b x^{n} + a\right ) + \log \left (\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{2 \, b n} \]

[In]

integrate(x^(-1+n)*arccsc(a+b*x^n),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {b {\left (\frac {2 \, {\left (b x^{n} + a\right )} \arcsin \left (\frac {1}{b x^{n} + a}\right )}{b^{2}} + \frac {\log \left (\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{b^{2}}\right )}}{2 \, n} \]

[In]

integrate(x^(-1+n)*arccsc(a+b*x^n),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{{\left (a+b\,x^n\right )}^2}}}\right )+\mathrm {asin}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]

[In]

int(x^(n - 1)*asin(1/(a + b*x^n)),x)

[Out]

(atanh(1/(1 - 1/(a + b*x^n)^2)^(1/2)) + asin(1/(a + b*x^n))*(a + b*x^n))/(b*n)