3.6.3 \(\int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{3/2}} \, dx\) [503]

3.6.3.1 Optimal result

Integrand size = 23, antiderivative size = 152 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {1+a^2 x^2}} \]

output

-1/2*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)+1/2*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*(a^2*x^2+1)^(1/2)*(a^2*c*x^2+c)^ 
(1/2)/a/arcsinh(a*x)^(1/2)
 

3.6.3.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {\sqrt {c+a^2 c x^2} \left (4+4 a^2 x^2+\sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{2 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]

input

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

output

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

3.6.3.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6205, 6195, 5971, 27, 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.

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

\(\Big \downarrow \) 6205

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

\(\Big \downarrow \) 6195

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {a^2 c x^2+c} \int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{a \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {a^2 c x^2+c} \int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a \sqrt {a^2 x^2+1}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \sqrt {a^2 c x^2+c} \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a \sqrt {a^2 x^2+1}}\)

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

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

output

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

3.6.3.3.1 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 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 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 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]
 

3.6.3.4 Maple [F]

\[\int \frac {\sqrt {a^{2} c \,x^{2}+c}}{\operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}}d x\]

input

int((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(3/2),x)
 

output

int((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(3/2),x)
 

3.6.3.5 Fricas [F(-2)]

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

input

integrate((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(3/2),x, algorithm="fricas")
 

output

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

3.6.3.6 Sympy [F]

\[ \int \frac {\sqrt {c+a^2 c x^2}}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

input

integrate((a**2*c*x**2+c)**(1/2)/asinh(a*x)**(3/2),x)
 

output

Integral(sqrt(c*(a**2*x**2 + 1))/asinh(a*x)**(3/2), x)
 

3.6.3.7 Maxima [F]

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

input

integrate((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(3/2),x, algorithm="maxima")
 

output

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

3.6.3.8 Giac [F]

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

input

integrate((a^2*c*x^2+c)^(1/2)/arcsinh(a*x)^(3/2),x, algorithm="giac")
 

output

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

3.6.3.9 Mupad [F(-1)]

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

input

int((c + a^2*c*x^2)^(1/2)/asinh(a*x)^(3/2),x)
 

output

int((c + a^2*c*x^2)^(1/2)/asinh(a*x)^(3/2), x)