\(\int x \text {arcsinh}(a x)^{3/2} \, dx\) [115]

Optimal result

Integrand size = 10, antiderivative size = 122 \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2} \] Output:

-3/8*x*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a+1/4*arcsinh(a*x)^(3/2)/a^2+1 
/2*x^2*arcsinh(a*x)^(3/2)-3/128*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arcsinh(a*x)^ 
(1/2))/a^2+3/128*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arcsinh(a*x)^(1/2))/a^2
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43 \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\Gamma \left (\frac {5}{2},2 \text {arcsinh}(a x)\right )}{16 \sqrt {2} a^2} \] Input:

Integrate[x*ArcSinh[a*x]^(3/2),x]
 

Output:

((Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -2*ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + Ga 
mma[5/2, 2*ArcSinh[a*x]])/(16*Sqrt[2]*a^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6192, 6227, 6195, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6198}

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.

\(\displaystyle \int x \text {arcsinh}(a x)^{3/2} \, dx\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {\text {arcsinh}(a x)}}dx}{4 a}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a^3}-\frac {\text {arcsinh}(a x)^{3/2}}{3 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\)

Input:

Int[x*ArcSinh[a*x]^(3/2),x]
 

Output:

(x^2*ArcSinh[a*x]^(3/2))/2 - (3*a*((x*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]] 
)/(2*a^2) - ArcSinh[a*x]^(3/2)/(3*a^3) + ((I/8)*((-1/2*I)*Sqrt[Pi/2]*Erf[S 
qrt[2]*Sqrt[ArcSinh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a* 
x]]]))/a^3))/4
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2611

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]
 

rule 2633

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]
 

rule 2634

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3789

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]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6192

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1))   Int 
[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free 
Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
 

rule 6195

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
1/(b*c^(m + 1))   Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, 
 a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-32 \operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+24 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -16 \operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }+3 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\right )-3 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\right )\right )}{128 \sqrt {\pi }\, a^{2}}\)	\(102\)

Input:

int(x*arcsinh(x*a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/128*2^(1/2)*(-32*arcsinh(x*a)^(3/2)*2^(1/2)*Pi^(1/2)*a^2*x^2+24*2^(1/2) 
*arcsinh(x*a)^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*a*x-16*arcsinh(x*a)^(3/2)*2 
^(1/2)*Pi^(1/2)+3*Pi*erf(2^(1/2)*arcsinh(x*a)^(1/2))-3*Pi*erfi(2^(1/2)*arc 
sinh(x*a)^(1/2)))/Pi^(1/2)/a^2
 

Fricas [F(-2)]

Exception generated. \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*arcsinh(a*x)^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int x \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \] Input:

integrate(x*asinh(a*x)**(3/2),x)
 

Output:

Integral(x*asinh(a*x)**(3/2), x)
 

Maxima [F]

\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \] Input:

integrate(x*arcsinh(a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate(x*arcsinh(a*x)^(3/2), x)
 

Giac [F]

\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \] Input:

integrate(x*arcsinh(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x*arcsinh(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \] Input:

int(x*asinh(a*x)^(3/2),x)
 

Output:

int(x*asinh(a*x)^(3/2), x)
 

Reduce [F]

\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int \sqrt {\mathit {asinh} \left (a x \right )}\, \mathit {asinh} \left (a x \right ) x d x \] Input:

int(x*asinh(a*x)^(3/2),x)
 

Output:

int(sqrt(asinh(a*x))*asinh(a*x)*x,x)