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

Optimal. Leaf size=128 \[ \frac{e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d}-\frac{e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.123552, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5863, 5657, 3307, 2181} \[ \frac{e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d}-\frac{e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c + d*x])^n,x]

[Out]

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

Rule 5863

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

Rule 5657

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

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]

Rubi steps

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

Mathematica [A]  time = 0.113041, size = 109, normalized size = 0.85 \[ \frac{e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )-e^{\frac{2 a}{b}} \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )^{-n} \text{Gamma}\left (n+1,\frac{a}{b}+\sinh ^{-1}(c+d x)\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*arcsinh(d*x+c))^n,x)

[Out]

int((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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsinh(d*x + c) + a)^n, 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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral((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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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