Integrand size = 8, antiderivative size = 94 \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\frac {15}{4} x \sqrt {\text {arcsinh}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{2 a}+x \text {arcsinh}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a} \] Output:
15/4*x*arcsinh(a*x)^(1/2)-5/2*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(3/2)/a+x*arc sinh(a*x)^(5/2)+15/16*Pi^(1/2)*erf(arcsinh(a*x)^(1/2))/a-15/16*Pi^(1/2)*er fi(arcsinh(a*x)^(1/2))/a
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.48 \[ \int \text {arcsinh}(a x)^{5/2} \, dx=-\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\Gamma \left (\frac {7}{2},\text {arcsinh}(a x)\right )}{2 a} \] Input:
Integrate[ArcSinh[a*x]^(5/2),x]
Output:
-1/2*((Sqrt[-ArcSinh[a*x]]*Gamma[7/2, -ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + Gamma[7/2, ArcSinh[a*x]])/a
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6187, 6213, 6187, 6234, 3042, 26, 3789, 2611, 2633, 2634}
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.
\(\displaystyle \int \text {arcsinh}(a x)^{5/2} \, dx\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \int \frac {x \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \int \sqrt {\text {arcsinh}(a x)}dx}{2 a}\right )\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}-\frac {1}{2} a \int \frac {x}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{2 a}\right )\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a x}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {i \sin (i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{2 a}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (i \int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{2 a}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{2 a}\right )}{2 a}\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle x \text {arcsinh}(a x)^{5/2}-\frac {5}{2} a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{2 a}\right )}{2 a}\right )\) |
Input:
Int[ArcSinh[a*x]^(5/2),x]
Output:
x*ArcSinh[a*x]^(5/2) - (5*a*((Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/a^2 - (3*(x*Sqrt[ArcSinh[a*x]] + ((I/2)*((-1/2*I)*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x] ]] + (I/2)*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]]))/a))/(2*a)))/2
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_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.91 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {-16 \operatorname {arcsinh}\left (x a \right )^{\frac {5}{2}} \sqrt {\pi }\, a x +40 \operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}-60 \sqrt {\operatorname {arcsinh}\left (x a \right )}\, \sqrt {\pi }\, a x -15 \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (x a \right )}\right )+15 \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (x a \right )}\right )}{16 \sqrt {\pi }\, a}\) | \(78\) |
Input:
int(arcsinh(x*a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/16*(-16*arcsinh(x*a)^(5/2)*Pi^(1/2)*a*x+40*arcsinh(x*a)^(3/2)*Pi^(1/2)* (a^2*x^2+1)^(1/2)-60*arcsinh(x*a)^(1/2)*Pi^(1/2)*a*x-15*Pi*erf(arcsinh(x*a )^(1/2))+15*Pi*erfi(arcsinh(x*a)^(1/2)))/Pi^(1/2)/a
Exception generated. \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(arcsinh(a*x)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \] Input:
integrate(asinh(a*x)**(5/2),x)
Output:
Integral(asinh(a*x)**(5/2), x)
\[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int { \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(arcsinh(a*x)^(5/2),x, algorithm="maxima")
Output:
integrate(arcsinh(a*x)^(5/2), x)
Exception generated. \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(arcsinh(a*x)^(5/2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \] Input:
int(asinh(a*x)^(5/2),x)
Output:
int(asinh(a*x)^(5/2), x)
\[ \int \text {arcsinh}(a x)^{5/2} \, dx=\int \sqrt {\mathit {asinh} \left (a x \right )}\, \mathit {asinh} \left (a x \right )^{2}d x \] Input:
int(asinh(a*x)^(5/2),x)
Output:
int(sqrt(asinh(a*x))*asinh(a*x)**2,x)