\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 (d+c^2 d x^2)} \, dx\) [244]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=-\frac {\frac {1}{12} i b^2 c^2 \pi ^3+\frac {a^2}{x^2}+\frac {2 a b c \sqrt {1+c^2 x^2}}{x}+\frac {2 a b \text {arcsinh}(c x)}{x^2}+\frac {2 b^2 c \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{x}+\frac {b^2 \text {arcsinh}(c x)^2}{x^2}-\frac {4}{3} b^2 c^2 \text {arcsinh}(c x)^3-2 b^2 c^2 \text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-4 a b c^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-4 a b c^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+4 a b c^2 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+2 b^2 c^2 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+2 a^2 c^2 \log (x)-2 b^2 c^2 \log (c x)-a^2 c^2 \log \left (1+c^2 x^2\right )+2 b^2 c^2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )-4 a b c^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-4 a b c^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )+2 a b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+2 b^2 c^2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+b^2 c^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )-b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \] Input:

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

Output:

-1/2*((I/12)*b^2*c^2*Pi^3 + a^2/x^2 + (2*a*b*c*Sqrt[1 + c^2*x^2])/x + (2*a 
*b*ArcSinh[c*x])/x^2 + (2*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/x + (b^2*A 
rcSinh[c*x]^2)/x^2 - (4*b^2*c^2*ArcSinh[c*x]^3)/3 - 2*b^2*c^2*ArcSinh[c*x] 
^2*Log[1 + E^(-2*ArcSinh[c*x])] - 4*a*b*c^2*ArcSinh[c*x]*Log[1 - I*E^ArcSi 
nh[c*x]] - 4*a*b*c^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 4*a*b*c^2*Ar 
cSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*b^2*c^2*ArcSinh[c*x]^2*Log[1 - 
E^(2*ArcSinh[c*x])] + 2*a^2*c^2*Log[x] - 2*b^2*c^2*Log[c*x] - a^2*c^2*Log[ 
1 + c^2*x^2] + 2*b^2*c^2*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - 4 
*a*b*c^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] - 4*a*b*c^2*PolyLog[2, I*E^ArcSin 
h[c*x]] + 2*a*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])] + 2*b^2*c^2*ArcSinh[c*x 
]*PolyLog[2, E^(2*ArcSinh[c*x])] + b^2*c^2*PolyLog[3, -E^(-2*ArcSinh[c*x]) 
] - b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/d
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.53 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6224, 27, 6214, 5984, 3042, 26, 4670, 3011, 2720, 6215, 14, 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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (c^2 d x^2+d\right )} \, dx\)

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*(a + b*ArcSinh[c*x])^2/(d*x^2) + (b*c*(-((Sqrt[1 + c^2*x^2]*(a + b*Ar 
cSinh[c*x]))/x) + b*c*Log[x]))/d - ((2*I)*c^2*(I*(a + b*ArcSinh[c*x])^2*Ar 
cTanh[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*ArcSinh[c*x])]) + (b*PolyLog[3, E^(2*A 
rcSinh[c*x])])/4)))/d
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 6215

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 + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), 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, m, p}, x] && EqQ 
[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 6224

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 + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 
1)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim 
p[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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
 

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. \(539\) vs. \(2(231)=462\).

Time = 1.61 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.78

method	result	size
derivativedivides	\(c^{2} \left (\frac {a^{2} \left (\frac {\ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{2 c^{2} x^{2}}-\ln \left (x c \right )\right )}{d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right ) \left (-2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 x^{2} c^{2}}+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )-2 \ln \left (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-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 (-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}-\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}\right )\)	\(540\)
default	\(c^{2} \left (\frac {a^{2} \left (\frac {\ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{2 c^{2} x^{2}}-\ln \left (x c \right )\right )}{d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right ) \left (-2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 x^{2} c^{2}}+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )-2 \ln \left (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-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 (-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}-\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}\right )\)	\(540\)
parts	\(\frac {a^{2} \left (-\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )+\frac {c^{2} \ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right ) \left (-2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 x^{2} c^{2}}+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )-2 \ln \left (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-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 \,c^{2} \left (-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}-\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}\)	\(543\)

Input:

int((a+b*arcsinh(x*c))^2/x^3/(c^2*d*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

c^2*(a^2/d*(1/2*ln(c^2*x^2+1)-1/2/c^2/x^2-ln(x*c))+1/d*b^2*(-1/2*arcsinh(x 
*c)*(-2*c^2*x^2+2*(c^2*x^2+1)^(1/2)*x*c+arcsinh(x*c))/x^2/c^2+ln(1+x*c+(c^ 
2*x^2+1)^(1/2))+ln(x*c+(c^2*x^2+1)^(1/2)-1)-2*ln(x*c+(c^2*x^2+1)^(1/2))-ar 
csinh(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*pol 
ylog(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)-1/2*polylog(3,-(x*c+ 
(c^2*x^2+1)^(1/2))^2))+2*a*b/d*(-1/2*((c^2*x^2+1)^(1/2)*x*c-c^2*x^2+arcsin 
h(x*c))/x^2/c^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))-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]

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

integrate((a+b*arcsinh(c*x))^2/x^3/(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^5 + d*x^ 
3), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx}{d} \] Input:

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

Output:

(Integral(a**2/(c**2*x**5 + x**3), x) + Integral(b**2*asinh(c*x)**2/(c**2* 
x**5 + x**3), x) + Integral(2*a*b*asinh(c*x)/(c**2*x**5 + x**3), x))/d
 

Maxima [F]

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

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

Output:

1/2*(c^2*log(c^2*x^2 + 1)/d - 2*c^2*log(x)/d - 1/(d*x^2))*a^2 + integrate( 
b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^5 + d*x^3) + 2*a*b*log(c*x + s 
qrt(c^2*x^2 + 1))/(c^2*d*x^5 + d*x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{2} x^{5}+x^{3}}d x \right ) a b \,x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{2} x^{5}+x^{3}}d x \right ) b^{2} x^{2}+\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2} c^{2} x^{2}-2 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}-a^{2}}{2 d \,x^{2}} \] Input:

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

Output:

(4*int(asinh(c*x)/(c**2*x**5 + x**3),x)*a*b*x**2 + 2*int(asinh(c*x)**2/(c* 
*2*x**5 + x**3),x)*b**2*x**2 + log(c**2*x**2 + 1)*a**2*c**2*x**2 - 2*log(x 
)*a**2*c**2*x**2 - a**2)/(2*d*x**2)