\(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx\) [216]

Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}-4 b c d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-2 b^2 c d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+2 b^2 c d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \] Output:

2*b^2*c^2*d*x-2*b*c*d*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))+2*c^2*d*x*(a+b* 
arcsinh(c*x))^2-d*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/x-4*b*c*d*(a+b*arcsinh( 
c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))-2*b^2*c*d*polylog(2,-c*x-(c^2*x^2+1)^ 
(1/2))+2*b^2*c*d*polylog(2,c*x+(c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {d \left (-a^2+a^2 c^2 x^2+2 a b c x \left (-\sqrt {1+c^2 x^2}+c x \text {arcsinh}(c x)\right )+b^2 c x \left (2 c x-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2\right )-2 a b \left (\text {arcsinh}(c x)+c x \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )\right )-b^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)+2 c x \left (-\log \left (1-e^{-\text {arcsinh}(c x)}\right )+\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )\right )-2 c x \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+2 c x \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )\right )}{x} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6222, 6187, 6213, 24, 6221, 24, 6231, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 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 6187

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcSinh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(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] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6221

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

rule 6222

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m 
 + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*( 
m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 + c^2*x 
^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

rule 6231

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

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.82

method	result	size
derivativedivides	\(c \left (a^{2} d \left (x c -\frac {1}{x c}\right )+b^{2} d \operatorname {arcsinh}\left (x c \right )^{2} x c -2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+2 x d c \,b^{2}-\frac {b^{2} d \operatorname {arcsinh}\left (x c \right )^{2}}{x c}+2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+2 a b d \left (x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(239\)
default	\(c \left (a^{2} d \left (x c -\frac {1}{x c}\right )+b^{2} d \operatorname {arcsinh}\left (x c \right )^{2} x c -2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+2 x d c \,b^{2}-\frac {b^{2} d \operatorname {arcsinh}\left (x c \right )^{2}}{x c}+2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+2 a b d \left (x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(239\)
parts	\(a^{2} d \left (c^{2} x -\frac {1}{x}\right )+b^{2} d \,c^{2} \operatorname {arcsinh}\left (x c \right )^{2} x -2 b^{2} d c \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+2 b^{2} c^{2} d x -\frac {b^{2} d \operatorname {arcsinh}\left (x c \right )^{2}}{x}+2 b^{2} d c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} c d \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} c d \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+2 a b d c \left (x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\)	\(243\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

b^2*c^2*d*x*arcsinh(c*x)^2 + 2*b^2*c^2*d*(x - sqrt(c^2*x^2 + 1)*arcsinh(c* 
x)/c) + a^2*c^2*d*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c*d - 2 
*(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*a*b*d - b^2*d*(log(c*x + sqrt( 
c^2*x^2 + 1))^2/x - integrate(2*(c^3*x^2 + sqrt(c^2*x^2 + 1)*c^2*x + c)*lo 
g(c*x + sqrt(c^2*x^2 + 1))/(c^3*x^4 + c*x^2 + (c^2*x^3 + x)*sqrt(c^2*x^2 + 
 1)), x)) - a^2*d/x
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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