Integrand size = 12, antiderivative size = 128 \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d}-\frac {e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{2 d} \]
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)
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85 \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\frac {e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-e^{\frac {2 a}{b}} \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )^{-n} \Gamma \left (1+n,\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{2 d} \]
((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))
Time = 0.41 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6273, 6189, 3042, 3788, 26, 2612}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a+b \text {arcsinh}(c+d x))^n \, dx\) |
\(\Big \downarrow \) 6273 |
\(\displaystyle \frac {\int (a+b \text {arcsinh}(c+d x))^nd(c+d x)}{d}\) |
\(\Big \downarrow \) 6189 |
\(\displaystyle \frac {\int (a+b \text {arcsinh}(c+d x))^n \cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (a+b \text {arcsinh}(c+d x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {\frac {1}{2} i \int -i e^{\frac {a-c-d x}{b}} (a+b \text {arcsinh}(c+d x))^nd(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int i e^{-\frac {a-c-d x}{b}} (a+b \text {arcsinh}(c+d x))^nd(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {1}{2} \int e^{-\frac {a-c-d x}{b}} (a+b \text {arcsinh}(c+d x))^nd(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \int e^{\frac {a-c-d x}{b}} (a+b \text {arcsinh}(c+d x))^nd(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle \frac {\frac {1}{2} b e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x))^n \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{2} b e^{a/b} (a+b \text {arcsinh}(c+d x))^n \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b d}\) |
((b*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)] )/(2*E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n) - (b*E^(a/b)*(a + b*ArcSin h[c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d*x])/b])/(2*((a + b*ArcSinh [c + d*x])/b)^n))/(b*d)
3.1.96.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{n}d x\]
\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} \,d x } \]
\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]
\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} \,d x } \]
\[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} \,d x } \]
Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^n \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]