\(\int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx\) [51]

Optimal result

Integrand size = 30, antiderivative size = 325 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}} \] Output:

(c^2*x^2+1)^(1/2)*(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*f^2+g^2)^(1/2)/(c^2*d*x^2+d)^(1/2)-(c^2*x^2+1)^( 
1/2)*(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*f^2+g^2)^(1/2)/(c^2*d*x^2+d)^(1/2)+b*(c^2*x^2+1)^(1/2)*polylo 
g(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))/(c^2*f^2+g^2)^(1 
/2)/(c^2*d*x^2+d)^(1/2)-b*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1 
/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/(c^2*f^2+g^2)^(1/2)/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx=\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {d} \left (g-c^2 f x\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}\right )}{\sqrt {d}}+\frac {b \sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x) \left (\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{\sqrt {d+c^2 d x^2}}}{\sqrt {c^2 f^2+g^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.73, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6260, 6258, 3042, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(2*Sqrt[1 + c^2*x^2]*((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*ArcSi 
nh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g + (b*P 
olyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/g))/(2*Sqrt[ 
c^2*f^2 + g^2])))/Sqrt[d + c^2*d*x^2]
 

Defintions of rubi rules used

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 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 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]
 

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {a \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \left (\ln \left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\ln \left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {dilog}\left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}\right )\)	\(529\)
parts	\(-\frac {a \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \left (\ln \left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\ln \left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {dilog}\left (\frac {-\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (x c +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}\right )\)	\(529\)

Input:

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

Output:

-a/g/(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)*(c^2*f^2+g^2)^ 
(1/2)*(c^2*x^2+1)^(1/2)/d/(c^4*f^2*x^2+c^2*g^2*x^2+c^2*f^2+g^2)*arcsinh(x* 
c)*(ln((-(x*c+(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)))-ln(((x*c+(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))))+(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)*(c^2*x^2+ 
1)^(1/2)/d/(c^4*f^2*x^2+c^2*g^2*x^2+c^2*f^2+g^2)*(dilog((-(x*c+(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)))-dilog(((x*c 
+(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)))) 
)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c**2*f**2 + g**2)*atan((sqrt(c**2*x**2 + 1)*g*i + c*f*i + c*g*i*x) 
/sqrt(c**2*f**2 + g**2))*a*i + int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*f + sqr 
t(c**2*x**2 + 1)*g*x),x)*b*c**2*f**2 + int(asinh(c*x)/(sqrt(c**2*x**2 + 1) 
*f + sqrt(c**2*x**2 + 1)*g*x),x)*b*g**2)/(sqrt(d)*(c**2*f**2 + g**2))