3.6.8 \(\int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx\) [508]

3.6.8.1 Optimal result

Integrand size = 23, antiderivative size = 182 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}} \]

output

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

3.6.8.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {c+a^2 c x^2} \left (1+a^2 x^2+4 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+\sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )}{3 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \]

input

Integrate[Sqrt[c + a^2*c*x^2]/ArcSinh[a*x]^(5/2),x]
 

output

(-2*Sqrt[c + a^2*c*x^2]*(1 + a^2*x^2 + 4*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x 
] + Sqrt[2]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] + Sqrt[2]*Ar 
cSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcSinh[a*x]]))/(3*a*Sqrt[1 + a^2*x^2]*ArcSi 
nh[a*x]^(3/2))
 

3.6.8.3 Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6205, 6193, 3042, 3788, 26, 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.

\(\displaystyle \int \frac {\sqrt {a^2 c x^2+c}}{\text {arcsinh}(a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6205

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

\(\Big \downarrow \) 6193

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

Int[Sqrt[c + a^2*c*x^2]/ArcSinh[a*x]^(5/2),x]
 

output

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

3.6.8.3.1 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 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 3788

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]
 

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]
 

rule 6205

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c^2*x^2]*(d + e*x^2)^p]*((a + b*ArcSinh[c*x] 
)^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x 
^2)^p/(1 + c^2*x^2)^p]   Int[x*(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] && LtQ[n, 
 -1]
 

3.6.8.4 Maple [F]

\[\int \frac {\sqrt {a^{2} c \,x^{2}+c}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]

input

int((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(5/2),x)
 

output

int((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(5/2),x)
 

3.6.8.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(5/2),x, algorithm="fricas")
 

output

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

3.6.8.6 Sympy [F]

\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

input

integrate((a**2*c*x**2+c)**(1/2)/asinh(a*x)**(5/2),x)
 

output

Integral(sqrt(c*(a**2*x**2 + 1))/asinh(a*x)**(5/2), x)
 

3.6.8.7 Maxima [F]

\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(5/2),x, algorithm="maxima")
 

output

integrate(sqrt(a^2*c*x^2 + c)/arcsinh(a*x)^(5/2), x)
 

3.6.8.8 Giac [F]

\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(5/2),x, algorithm="giac")
 

output

integrate(sqrt(a^2*c*x^2 + c)/arcsinh(a*x)^(5/2), x)
 

3.6.8.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {c\,a^2\,x^2+c}}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]

input

int((c + a^2*c*x^2)^(1/2)/asinh(a*x)^(5/2),x)
 

output

int((c + a^2*c*x^2)^(1/2)/asinh(a*x)^(5/2), x)