3.1.95 \(\int x (a+b \sinh ^{-1}(c+d x))^n \, dx\) [95]

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

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 267, normalized size of antiderivative = 1.00, 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 = {5859, 5830, 6873, 12, 6874, 3388, 2212, 5556, 3389} \begin {gather*} \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} \end {gather*}

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 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 3388

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 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 5830

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 5859

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 6873

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

Rule 6874

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

Rubi steps

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

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Mathematica [A]
time = 0.96, size = 330, normalized size = 1.24 \begin {gather*} \frac {2^{-3-n} \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 (-2^{2+n} c \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )^n \cosh \left (\frac {a}{b}\right ) \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )+\left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^n \cosh \left (\frac {2 a}{b}\right ) \Gamma \left (1+n,\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+2^{2+n} c \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )^n \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )+2^{2+n} c \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\sinh ^{-1}(c+d x)\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )^n \Gamma \left (1+n,-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+\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 ) \sinh \left (\frac {2 a}{b}\right )\right )}{d^2} \end {gather*}

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*(a/b + ArcSinh[c + d*x])^n*Cosh[a/b]*Gamma[1 + n, -((a +
b*ArcSinh[c + d*x])/b)]) + (-((a + b*ArcSinh[c + d*x])/b))^n*Cosh[(2*a)/b]*Gamma[1 + n, (2*(a + b*ArcSinh[c +
d*x]))/b] + 2^(2 + n)*c*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)]*Sinh[a/b] + 2^
(2 + n)*c*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, a/b + ArcSinh[c + d*x]]*(Cosh[a/b] + Sinh[a/b]) + (a/
b + ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + (-((a
+ b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b]))/(d^2*(-((a + b*ArcSin
h[c + d*x])^2/b^2))^n)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x \left (a +b \arcsinh \left (d x +c \right )\right )^{n}\, dx\]

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 = 0.00, 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*(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 = 0.00, 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*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \end {gather*}

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 = 0.00, 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*(a+b*arcsinh(d*x+c))^n,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asinh(c + d*x))^n,x)

[Out]

int(x*(a + b*asinh(c + d*x))^n, x)

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