3.1.96 \(\int (a+b \sinh ^{-1}(c+d x))^n \, dx\) [96]

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

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, 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 = {5858, 5774, 3388, 2212} \begin {gather*} \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} \end {gather*}

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 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 5774

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 5858

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 138, normalized size = 1.08 \begin {gather*} \frac {\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 \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\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 )\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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 = 0.00, 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((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 = 0.00, 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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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((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 = 0.00, 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((a+b*arcsinh(d*x+c))^n,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\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((a + b*asinh(c + d*x))^n,x)

[Out]

int((a + b*asinh(c + d*x))^n, x)

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