\(\int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx\) [131]

Optimal result

Integrand size = 12, antiderivative size = 188 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 x^4 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5} \] Output:

-2*x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(1/2)-1/8*Pi^(1/2)*erf(arcsinh(a*x 
)^(1/2))/a^5+3/16*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arcsinh(a*x)^(1/2))/a^5-1/1 
6*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*arcsinh(a*x)^(1/2))/a^5+1/8*Pi^(1/2)*erfi(a 
rcsinh(a*x)^(1/2))/a^5-3/16*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*arcsinh(a*x)^(1/ 
2))/a^5+1/16*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*arcsinh(a*x)^(1/2))/a^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.41 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {e^{-5 \text {arcsinh}(a x)} \left (-1+3 e^{2 \text {arcsinh}(a x)}-2 e^{4 \text {arcsinh}(a x)}-2 e^{6 \text {arcsinh}(a x)}+3 e^{8 \text {arcsinh}(a x)}-e^{10 \text {arcsinh}(a x)}+\sqrt {5} e^{5 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-5 \text {arcsinh}(a x)\right )-3 \sqrt {3} e^{5 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )+2 e^{5 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+2 e^{5 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )-3 \sqrt {3} e^{5 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )+\sqrt {5} e^{5 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},5 \text {arcsinh}(a x)\right )\right )}{16 a^5 \sqrt {\text {arcsinh}(a x)}} \] Input:

Integrate[x^4/ArcSinh[a*x]^(3/2),x]
 

Output:

(-1 + 3*E^(2*ArcSinh[a*x]) - 2*E^(4*ArcSinh[a*x]) - 2*E^(6*ArcSinh[a*x]) + 
 3*E^(8*ArcSinh[a*x]) - E^(10*ArcSinh[a*x]) + Sqrt[5]*E^(5*ArcSinh[a*x])*S 
qrt[-ArcSinh[a*x]]*Gamma[1/2, -5*ArcSinh[a*x]] - 3*Sqrt[3]*E^(5*ArcSinh[a* 
x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -3*ArcSinh[a*x]] + 2*E^(5*ArcSinh[a*x]) 
*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]] + 2*E^(5*ArcSinh[a*x])*Sqrt 
[ArcSinh[a*x]]*Gamma[1/2, ArcSinh[a*x]] - 3*Sqrt[3]*E^(5*ArcSinh[a*x])*Sqr 
t[ArcSinh[a*x]]*Gamma[1/2, 3*ArcSinh[a*x]] + Sqrt[5]*E^(5*ArcSinh[a*x])*Sq 
rt[ArcSinh[a*x]]*Gamma[1/2, 5*ArcSinh[a*x]])/(16*a^5*E^(5*ArcSinh[a*x])*Sq 
rt[ArcSinh[a*x]])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6193, 2009}

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.

\(\displaystyle \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 6193

\(\displaystyle \frac {2 \int \left (\frac {a x}{8 \sqrt {\text {arcsinh}(a x)}}-\frac {9 \sinh (3 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {5 \sinh (5 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^5}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {3}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\)

Input:

Int[x^4/ArcSinh[a*x]^(3/2),x]
 

Output:

(-2*x^4*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) + (2*(-1/16*(Sqrt[Pi]*Er 
f[Sqrt[ArcSinh[a*x]]]) + (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/32 
 - (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi]*Erfi[Sqrt[A 
rcSinh[a*x]]])/16 - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/32 + ( 
Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/32))/a^5
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6193

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si 
mp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- 
a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi 
nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 
1]
 

Maple [F]

\[\int \frac {x^{4}}{\operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}}}d x\]

Input:

int(x^4/arcsinh(x*a)^(3/2),x)
 

Output:

int(x^4/arcsinh(x*a)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/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]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/asinh(a*x)**(3/2),x)
 

Output:

Integral(x**4/asinh(a*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4/arcsinh(a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^4/arcsinh(a*x)^(3/2), x)
 

Giac [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4/arcsinh(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^4/arcsinh(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(x^4/asinh(a*x)^(3/2),x)
 

Output:

int(x^4/asinh(a*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {\frac {4 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asinh} \left (a x \right )}}{\mathit {asinh} \left (a x \right )^{2} a^{2} x^{2}+\mathit {asinh} \left (a x \right )^{2}}d x \right ) a}{3}+\frac {4 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asinh} \left (a x \right )}\, x^{2}}{\mathit {asinh} \left (a x \right )^{2} a^{2} x^{2}+\mathit {asinh} \left (a x \right )^{2}}d x \right ) a^{3}}{3}+10 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{5}}{\mathit {asinh} \left (a x \right ) a^{2} x^{2}+\mathit {asinh} \left (a x \right )}d x \right ) a^{6}+8 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{3}}{\mathit {asinh} \left (a x \right ) a^{2} x^{2}+\mathit {asinh} \left (a x \right )}d x \right ) a^{4}-\frac {8 \mathit {asinh} \left (a x \right ) \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x}{\mathit {asinh} \left (a x \right ) a^{2} x^{2}+\mathit {asinh} \left (a x \right )}d x \right ) a^{2}}{3}-2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, a^{4} x^{4}+\frac {8 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}}{3}}{\mathit {asinh} \left (a x \right ) a^{5}} \] Input:

int(x^4/asinh(a*x)^(3/2),x)
 

Output:

(2*(2*asinh(a*x)*int(sqrt(asinh(a*x))/(asinh(a*x)**2*a**2*x**2 + asinh(a*x 
)**2),x)*a + 2*asinh(a*x)*int((sqrt(asinh(a*x))*x**2)/(asinh(a*x)**2*a**2* 
x**2 + asinh(a*x)**2),x)*a**3 + 15*asinh(a*x)*int((sqrt(a**2*x**2 + 1)*sqr 
t(asinh(a*x))*x**5)/(asinh(a*x)*a**2*x**2 + asinh(a*x)),x)*a**6 + 12*asinh 
(a*x)*int((sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x))*x**3)/(asinh(a*x)*a**2*x** 
2 + asinh(a*x)),x)*a**4 - 4*asinh(a*x)*int((sqrt(a**2*x**2 + 1)*sqrt(asinh 
(a*x))*x)/(asinh(a*x)*a**2*x**2 + asinh(a*x)),x)*a**2 - 3*sqrt(a**2*x**2 + 
 1)*sqrt(asinh(a*x))*a**4*x**4 + 4*sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x))))/ 
(3*asinh(a*x)*a**5)