3.95 \(\int x (a+b \sinh ^{-1}(c+d x))^n \, dx\)

Optimal. Leaf size=267 \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^2}-\frac{c e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^2}+\frac{c e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^2}+\frac{2^{-n-3} e^{\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^2} \]

[Out]

(2^(-3 - n)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b])/(d^2*E^((2*a)/b)*(-((a +
b*ArcSinh[c + d*x])/b))^n) - (c*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(2*d^
2*E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n) + (c*E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSi
nh[c + d*x])/b])/(2*d^2*((a + b*ArcSinh[c + d*x])/b)^n) + (2^(-3 - n)*E^((2*a)/b)*(a + b*ArcSinh[c + d*x])^n*G
amma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b])/(d^2*((a + b*ArcSinh[c + d*x])/b)^n)

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Rubi [A]  time = 0.483293, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5865, 5805, 6741, 12, 6742, 3307, 2181, 5448, 3308} \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^2}-\frac{c e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^2}+\frac{c e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^2}+\frac{2^{-n-3} e^{\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSinh[c + d*x])^n,x]

[Out]

(2^(-3 - n)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b])/(d^2*E^((2*a)/b)*(-((a +
b*ArcSinh[c + d*x])/b))^n) - (c*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(2*d^
2*E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n) + (c*E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSi
nh[c + d*x])/b])/(2*d^2*((a + b*ArcSinh[c + d*x])/b)^n) + (2^(-3 - n)*E^((2*a)/b)*(a + b*ArcSinh[c + d*x])^n*G
amma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b])/(d^2*((a + b*ArcSinh[c + d*x])/b)^n)

Rule 5865

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

Rule 5805

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3307

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

Rule 2181

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

Rule 5448

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 3308

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]

Rubi steps

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

Mathematica [A]  time = 0.179955, size = 228, normalized size = 0.85 \[ \frac{2^{-n-3} e^{-\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-c 2^{n+2} e^{a/b} \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )^n \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )+c 2^{n+2} e^{\frac{3 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+e^{\frac{4 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSinh[c + d*x])^n,x]

[Out]

(2^(-3 - n)*(a + b*ArcSinh[c + d*x])^n*(2^(2 + n)*c*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n,
a/b + ArcSinh[c + d*x]] + (a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b] - 2^(2 + n
)*c*E^(a/b)*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)] + E^((4*a)/b)*(-((a + b*Arc
Sinh[c + d*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b]))/(d^2*E^((2*a)/b)*(-((a + b*ArcSinh[c + d*x
])^2/b^2))^n)

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Maple [F]  time = 0.151, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsinh(d*x+c))^n,x)

[Out]

int(x*(a+b*arcsinh(d*x+c))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*arcsinh(d*x + c) + a)^n*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*arcsinh(d*x + c) + a)^n*x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asinh(d*x+c))**n,x)

[Out]

Integral(x*(a + b*asinh(c + d*x))**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^n*x, x)