\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x (d+c^2 d x^2)} \, dx\) [242]

Optimal result

Integrand size = 26, antiderivative size = 116 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \] Output:

-2*(a+b*arcsinh(c*x))^2*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d-b*(a+b*arcsin 
h(c*x))*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/d+b*(a+b*arcsinh(c*x))*polyl 
og(2,(c*x+(c^2*x^2+1)^(1/2))^2)/d+1/2*b^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2 
))^2)/d-1/2*b^2*polylog(3,(c*x+(c^2*x^2+1)^(1/2))^2)/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(400\) vs. \(2(116)=232\).

Time = 0.25 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.45 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 a^3+6 a^2 b \text {arcsinh}(c x)+12 a b^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+6 b^3 \text {arcsinh}(c x)^2 \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+12 a b^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+6 b^3 \text {arcsinh}(c x)^2 \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-6 a^2 b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-12 a b^2 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-6 b^3 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3 a^2 b \log \left (1+c^2 x^2\right )+12 b^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+12 b^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-6 a b^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-6 b^3 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-12 b^3 \operatorname {PolyLog}\left (3,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-12 b^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+3 b^3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{6 b d} \] Input:

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

Output:

-1/6*(2*a^3 + 6*a^2*b*ArcSinh[c*x] + 12*a*b^2*ArcSinh[c*x]*Log[1 + (c*E^Ar 
cSinh[c*x])/Sqrt[-c^2]] + 6*b^3*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/ 
Sqrt[-c^2]] + 12*a*b^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] 
 + 6*b^3*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 6*a^2*b*L 
og[1 - E^(2*ArcSinh[c*x])] - 12*a*b^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c* 
x])] - 6*b^3*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 3*a^2*b*Log[1 + 
c^2*x^2] + 12*b^2*(a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[ 
-c^2]] + 12*b^2*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x] 
)/c] - 6*a*b^2*PolyLog[2, E^(2*ArcSinh[c*x])] - 6*b^3*ArcSinh[c*x]*PolyLog 
[2, E^(2*ArcSinh[c*x])] - 12*b^3*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] 
 - 12*b^3*PolyLog[3, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 3*b^3*PolyLog[3, E^( 
2*ArcSinh[c*x])])/(b*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6214, 5984, 3042, 26, 4670, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

\(\displaystyle \frac {2 \int (a+b \text {arcsinh}(c x))^2 \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \int i (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 i \int (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

\(\displaystyle \frac {2 i \left (-i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {2 i \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )}{d}\)

Input:

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

Output:

((2*I)*(I*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])] - I*b*(-1/2*( 
(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])]) + (b*PolyLog[3, -E^( 
2*ArcSinh[c*x])])/4) + I*b*(-1/2*((a + b*ArcSinh[c*x])*PolyLog[2, E^(2*Arc 
Sinh[c*x])]) + (b*PolyLog[3, E^(2*ArcSinh[c*x])])/4)))/d
 

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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 6214

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[1/d   Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar 
cSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs. \(2(157)=314\).

Time = 1.30 (sec) , antiderivative size = 392, normalized size of antiderivative = 3.38

method	result	size
parts	\(\frac {a^{2} \left (\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b^{2} \left (\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}\)	\(392\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (x c \right )\right )}{d}+\frac {b^{2} \left (\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}\)	\(394\)
default	\(\frac {a^{2} \left (-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (x c \right )\right )}{d}+\frac {b^{2} \left (\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}\)	\(394\)

Input:

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

Output:

a^2/d*(ln(x)-1/2*ln(c^2*x^2+1))+1/d*b^2*(arcsinh(x*c)^2*ln(1+x*c+(c^2*x^2+ 
1)^(1/2))+2*arcsinh(x*c)*polylog(2,-x*c-(c^2*x^2+1)^(1/2))-2*polylog(3,-x* 
c-(c^2*x^2+1)^(1/2))+arcsinh(x*c)^2*ln(1-x*c-(c^2*x^2+1)^(1/2))+2*arcsinh( 
x*c)*polylog(2,x*c+(c^2*x^2+1)^(1/2))-2*polylog(3,x*c+(c^2*x^2+1)^(1/2))-a 
rcsinh(x*c)^2*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-arcsinh(x*c)*polylog(2,-(x*c 
+(c^2*x^2+1)^(1/2))^2)+1/2*polylog(3,-(x*c+(c^2*x^2+1)^(1/2))^2))+2*a*b/d* 
(arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))+polylog(2,-x*c-(c^2*x^2+1)^(1/2) 
)+arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))+polylog(2,x*c+(c^2*x^2+1)^(1/2) 
)-arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-1/2*polylog(2,-(x*c+(c^2*x^ 
2+1)^(1/2))^2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

(Integral(a**2/(c**2*x**3 + x), x) + Integral(b**2*asinh(c*x)**2/(c**2*x** 
3 + x), x) + Integral(2*a*b*asinh(c*x)/(c**2*x**3 + x), x))/d
 

Maxima [F]

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

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

Output:

-1/2*a^2*(log(c^2*x^2 + 1)/d - 2*log(x)/d) + integrate(b^2*log(c*x + sqrt( 
c^2*x^2 + 1))^2/(c^2*d*x^3 + d*x) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^ 
2*d*x^3 + d*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )} \, dx=\frac {4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{2} x^{3}+x}d x \right ) a b +2 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{2} x^{3}+x}d x \right ) b^{2}-\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2}+2 \,\mathrm {log}\left (x \right ) a^{2}}{2 d} \] Input:

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

Output:

(4*int(asinh(c*x)/(c**2*x**3 + x),x)*a*b + 2*int(asinh(c*x)**2/(c**2*x**3 
+ x),x)*b**2 - log(c**2*x**2 + 1)*a**2 + 2*log(x)*a**2)/(2*d)