3.1.52 \(\int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx\) [52]

3.1.52.1 Optimal result

Integrand size = 30, antiderivative size = 444 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=-\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \]

output

-g*(c^2*x^2+1)*(a+b*arcsinh(c*x))/(c^2*f^2+g^2)/(g*x+f)/(c^2*d*x^2+d)^(1/2 
)+b*c*ln(g*x+f)*(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)/(c^2*d*x^2+d)^(1/2)+c^2*f* 
(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2) 
))*(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)^(3/2)/(c^2*d*x^2+d)^(1/2)-c^2*f*(a+b*ar 
csinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2 
*x^2+1)^(1/2)/(c^2*f^2+g^2)^(3/2)/(c^2*d*x^2+d)^(1/2)+b*c^2*f*polylog(2,-( 
c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*x^2+1)^(1/2)/(c^2 
*f^2+g^2)^(3/2)/(c^2*d*x^2+d)^(1/2)-b*c^2*f*polylog(2,-(c*x+(c^2*x^2+1)^(1 
/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)^(3/2)/(c 
^2*d*x^2+d)^(1/2)
 

3.1.52.2 Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\frac {c \sqrt {1+c^2 x^2} \left (-\frac {g \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c f+c g x}+b \log (f+g x)+\frac {c f \left ((a+b \text {arcsinh}(c x)) \left (\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}} \]

input

Integrate[(a + b*ArcSinh[c*x])/((f + g*x)^2*Sqrt[d + c^2*d*x^2]),x]
 

output

(c*Sqrt[1 + c^2*x^2]*(-((g*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c*f + 
c*g*x)) + b*Log[f + g*x] + (c*f*((a + b*ArcSinh[c*x])*(Log[1 + (E^ArcSinh[ 
c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] - Log[1 + (E^ArcSinh[c*x]*g)/(c*f + S 
qrt[c^2*f^2 + g^2])]) + b*PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2 
*f^2 + g^2])] - b*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^ 
2]))]))/Sqrt[c^2*f^2 + g^2]))/((c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])
 

3.1.52.3 Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.74, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {6260, 6258, 3042, 3805, 3042, 3147, 16, 3803, 2694, 27, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d} (f+g x)^2} \, dx\)

\(\Big \downarrow \) 6260

\(\displaystyle \frac {\sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6258

\(\displaystyle \frac {c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{(c f+c g x)^2}d\text {arcsinh}(c x)}{\sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c \sqrt {c^2 x^2+1} \left (\frac {c f \int \frac {a+b \text {arcsinh}(c x)}{c f-i g \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 f^2+g^2}+\frac {b g \int \frac {\cos (i \text {arcsinh}(c x))}{c f-i g \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 f^2+g^2}-\frac {g \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (c f+c g x)}\right )}{\sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 3147

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

\(\Big \downarrow \) 16

\(\displaystyle \frac {c \sqrt {c^2 x^2+1} \left (\frac {c f \int \frac {a+b \text {arcsinh}(c x)}{c f-i g \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 f^2+g^2}-\frac {g \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2+g^2}\right )}{\sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {c \sqrt {c^2 x^2+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )de^{\text {arcsinh}(c x)}}{g}\right )}{2 \sqrt {c^2 f^2+g^2}}-\frac {g \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )de^{\text {arcsinh}(c x)}}{g}\right )}{2 \sqrt {c^2 f^2+g^2}}\right )}{c^2 f^2+g^2}-\frac {g \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2+g^2}\right )}{\sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {c \sqrt {c^2 x^2+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2+g^2}}-\frac {g \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2+g^2}}\right )}{c^2 f^2+g^2}-\frac {g \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2+g^2}\right )}{\sqrt {c^2 d x^2+d}}\)

input

Int[(a + b*ArcSinh[c*x])/((f + g*x)^2*Sqrt[d + c^2*d*x^2]),x]
 

output

(c*Sqrt[1 + c^2*x^2]*(-((g*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/((c^2*f 
^2 + g^2)*(c*f + c*g*x))) + (b*Log[c*f + c*g*x])/(c^2*f^2 + g^2) + (2*c*f* 
((g*(((a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 
+ g^2])])/g + (b*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2 
]))])/g))/(2*Sqrt[c^2*f^2 + g^2]) - (g*(((a + b*ArcSinh[c*x])*Log[1 + (E^A 
rcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g + (b*PolyLog[2, -((E^ArcSin 
h[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/g))/(2*Sqrt[c^2*f^2 + g^2])))/(c^ 
2*f^2 + g^2)))/Sqrt[d + c^2*d*x^2]
 

3.1.52.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( 
-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; 
FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 6258

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S 
qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[I 
nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b 
, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt 
Q[m, 0] || IGtQ[n, 0])
 

rule 6260

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

3.1.52.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1769\) vs. \(2(442)=884\).

Time = 0.85 (sec) , antiderivative size = 1770, normalized size of antiderivative = 3.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1770\)
parts	\(\text {Expression too large to display}\)	\(1770\)

input

int((a+b*arcsinh(c*x))/(g*x+f)^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBO 
SE)
 

output

-a/d/(c^2*f^2+g^2)/(x+f/g)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2 
+g^2)/g^2)^(1/2)-a/g*c^2*f/(c^2*f^2+g^2)/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2 
*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x 
+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))+b*( 
d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)/(c^2*f^2+g^2)/(g*x+f)*x^3* 
c^4*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*f^2+g^2)/(g*x+f)*x*c^2*f 
-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)/(c^2*f^2+g^2)/(g*x+f)* 
x^2*c^2*g+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)^(1/2)/(c^2*f^ 
2+g^2)/(g*x+f)*x*c*g+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)/(c 
^2*f^2+g^2)/(g*x+f)*x*c^2*f+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^ 
2+1)^(1/2)/(c^2*f^2+g^2)/(g*x+f)*c*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/ 
d/(c^2*x^2+1)/(c^2*f^2+g^2)/(g*x+f)*g+b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^ 
(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g 
^4)*c^3*ln((c*x+(c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*x+(c^2*x^2+1)^(1/2))-g)*f^ 
2-2*b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2 
*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c^3*ln(c*x+(c^2*x^2+1)^(1/2))* 
f^2+b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2 
*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c^2*ln((-(c*x+(c^2*x^2+1)^(1/2 
))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)*(c^ 
2*f^2+g^2)^(1/2)*f-b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4...
 

3.1.52.5 Fricas [F]

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

input

integrate((a+b*arcsinh(c*x))/(g*x+f)^2/(c^2*d*x^2+d)^(1/2),x, algorithm="f 
ricas")
 

output

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^2*d*g^2*x^4 + 2*c^2*d 
*f*g*x^3 + 2*d*f*g*x + d*f^2 + (c^2*d*f^2 + d*g^2)*x^2), x)
 

3.1.52.6 Sympy [F]

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

input

integrate((a+b*asinh(c*x))/(g*x+f)**2/(c**2*d*x**2+d)**(1/2),x)
 

output

Integral((a + b*asinh(c*x))/(sqrt(d*(c**2*x**2 + 1))*(f + g*x)**2), x)
 

3.1.52.7 Maxima [F]

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

input

integrate((a+b*arcsinh(c*x))/(g*x+f)^2/(c^2*d*x^2+d)^(1/2),x, algorithm="m 
axima")
 

output

integrate((b*arcsinh(c*x) + a)/(sqrt(c^2*d*x^2 + d)*(g*x + f)^2), x)
 

3.1.52.8 Giac [F]

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

input

integrate((a+b*arcsinh(c*x))/(g*x+f)^2/(c^2*d*x^2+d)^(1/2),x, algorithm="g 
iac")
 

output

integrate((b*arcsinh(c*x) + a)/(sqrt(c^2*d*x^2 + d)*(g*x + f)^2), x)
 

3.1.52.9 Mupad [F(-1)]

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

input

int((a + b*asinh(c*x))/((f + g*x)^2*(d + c^2*d*x^2)^(1/2)),x)
 

output

int((a + b*asinh(c*x))/((f + g*x)^2*(d + c^2*d*x^2)^(1/2)), x)