\(\int (a+b \text {arcsinh}(c+d x))^n \, dx\) [96]

Optimal result

Integrand size = 12, antiderivative size = 128 \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d}-\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d} \]

[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)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5858, 5774, 3388, 2212} \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d}-\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d} \]

[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*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b \text {arcsinh}(x))^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 \text {arcsinh}(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 \text {arcsinh}(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 \text {arcsinh}(c+d x)\right )}{2 b d} \\ & = \frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d}-\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85 \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-e^{\frac {2 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^{-n} \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{2 d} \]

[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))

Maple [F]

\[\int \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{n}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]

[In]

int((a + b*asinh(c + d*x))^n,x)

[Out]

int((a + b*asinh(c + d*x))^n, x)