3.2.32 \(\int x \sinh ^{-1}(a x)^n \, dx\) [132]

Optimal. Leaf size=59 \[ \frac {2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-3-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^2} \]

[Out]

2^(-3-n)*arcsinh(a*x)^n*GAMMA(1+n,-2*arcsinh(a*x))/a^2/((-arcsinh(a*x))^n)+2^(-3-n)*GAMMA(1+n,2*arcsinh(a*x))/
a^2

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Rubi [A]
time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5780, 5556, 12, 3389, 2212} \begin {gather*} \frac {2^{-n-3} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \text {Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcSinh[a*x]^n,x]

[Out]

(2^(-3 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]])/(a^2*(-ArcSinh[a*x])^n) + (2^(-3 - n)*Gamma[1 + n, 2
*ArcSinh[a*x]])/a^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int x \sinh ^{-1}(a x)^n \, dx &=\frac {\text {Subst}\left (\int x^n \cosh (x) \sinh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int \frac {1}{2} x^n \sinh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^2}+\frac {\text {Subst}\left (\int e^{2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-3-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.00 \begin {gather*} \frac {2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-3-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSinh[a*x]^n,x]

[Out]

(2^(-3 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]])/(a^2*(-ArcSinh[a*x])^n) + (2^(-3 - n)*Gamma[1 + n, 2
*ArcSinh[a*x]])/a^2

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 1.71, size = 38, normalized size = 0.64

method	result	size
default	\(\frac {\arcsinh \left (a x \right )^{n +2} \hypergeom \left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \arcsinh \left (a x \right )^{2}\right )}{a^{2} \left (n +2\right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsinh(a*x)^n,x,method=_RETURNVERBOSE)

[Out]

1/a^2/(n+2)*arcsinh(a*x)^(n+2)*hypergeom([1+1/2*n],[3/2,2+1/2*n],arcsinh(a*x)^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)^n,x, algorithm="maxima")

[Out]

integrate(x*arcsinh(a*x)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)^n,x, algorithm="fricas")

[Out]

integral(x*arcsinh(a*x)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \operatorname {asinh}^{n}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asinh(a*x)**n,x)

[Out]

Integral(x*asinh(a*x)**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)^n,x, algorithm="giac")

[Out]

integrate(x*arcsinh(a*x)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asinh(a*x)^n,x)

[Out]

int(x*asinh(a*x)^n, x)

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