\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx\) [306]

Optimal result

Integrand size = 28, antiderivative size = 293 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}+\frac {2 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {4 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}} \] Output:

-1/3*b^2*c^2*(c^2*d*x^2+d)^(1/2)/d/x-1/3*b*c*(c^2*x^2+1)^(1/2)*(a+b*arcsin 
h(c*x))/x^2/(c^2*d*x^2+d)^(1/2)+2/3*c^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x 
))^2/(c^2*d*x^2+d)^(1/2)-1/3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/d/x^ 
3+2/3*c^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/d/x-4/3*b*c^3*(c^2*x^2+ 
1)^(1/2)*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))^2)/(c^2*d*x^2+d)^ 
(1/2)-2/3*b^2*c^3*(c^2*x^2+1)^(1/2)*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)/( 
c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\frac {-a^2+a^2 c^2 x^2-b^2 c^2 x^2+2 a^2 c^4 x^4-b^2 c^4 x^4-a b c x \sqrt {1+c^2 x^2}+b^2 \left (-1+c^2 x^2+2 c^4 x^4-2 c^3 x^3 \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2-b \text {arcsinh}(c x) \left (b c x \sqrt {1+c^2 x^2}-2 a \left (-1+c^2 x^2+2 c^4 x^4\right )+4 b c^3 x^3 \sqrt {1+c^2 x^2} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )-4 a b c^3 x^3 \sqrt {1+c^2 x^2} \log (c x)+2 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 x^3 \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/(x^4*Sqrt[d + c^2*d*x^2]),x]
 

Output:

(-a^2 + a^2*c^2*x^2 - b^2*c^2*x^2 + 2*a^2*c^4*x^4 - b^2*c^4*x^4 - a*b*c*x* 
Sqrt[1 + c^2*x^2] + b^2*(-1 + c^2*x^2 + 2*c^4*x^4 - 2*c^3*x^3*Sqrt[1 + c^2 
*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]*(b*c*x*Sqrt[1 + c^2*x^2] - 2*a*(-1 
+ c^2*x^2 + 2*c^4*x^4) + 4*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[1 - E^(-2*ArcSi 
nh[c*x])]) - 4*a*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[c*x] + 2*b^2*c^3*x^3*Sqrt 
[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*x^3*Sqrt[d + c^2*d*x^2])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.90, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6224, 6191, 242, 6215, 6190, 25, 3042, 26, 4201, 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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {c^2 d x^2+d}} \, dx\)

\(\Big \downarrow \) 6224

\(\displaystyle -\frac {2}{3} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {c^2 d x^2+d}}dx+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x^3}dx}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6191

\(\displaystyle -\frac {2}{3} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {c^2 d x^2+d}}dx+\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {1}{2} b c \int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {2}{3} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {c^2 d x^2+d}}dx+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6215

\(\displaystyle -\frac {2}{3} c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x}dx}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 6190

\(\displaystyle -\frac {2}{3} c^2 \left (\frac {2 c \sqrt {c^2 x^2+1} \int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2}{3} c^2 \left (-\frac {2 c \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2}{3} c^2 \left (-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c \sqrt {c^2 x^2+1} \int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2}{3} c^2 \left (-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {2}{3} c^2 \left (-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {2}{3} c^2 \left (-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {2}{3} c^2 \left (-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}\)

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcSinh[c*x])^2/(x^4*Sqrt[d + c^2*d*x^2]),x]
 

Output:

-1/3*(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(d*x^3) + (2*b*c*Sqrt[1 
+ c^2*x^2]*(-1/2*(b*c*Sqrt[1 + c^2*x^2])/x - (a + b*ArcSinh[c*x])/(2*x^2)) 
)/(3*Sqrt[d + c^2*d*x^2]) - (2*c^2*(-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[ 
c*x])^2)/(d*x)) + ((2*I)*c*Sqrt[1 + c^2*x^2]*((-1/2*I)*(a + b*ArcSinh[c*x] 
)^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c*x])*Log[1 + E^((2*a)/b - I*Pi - (2*( 
a + b*ArcSinh[c*x]))/b)]) + (b^2*PolyLog[2, -a - b*ArcSinh[c*x]])/4)))/Sqr 
t[d + c^2*d*x^2]))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

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 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 4201

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

rule 6190

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

rule 6191

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1504\) vs. \(2(275)=550\).

Time = 1.27 (sec) , antiderivative size = 1505, normalized size of antiderivative = 5.14

method	result	size
default	\(\text {Expression too large to display}\)	\(1505\)
parts	\(\text {Expression too large to display}\)	\(1505\)

Input:

int((a+b*arcsinh(x*c))^2/x^4/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d*x*(c^2*x^2+1)*arcs 
inh(x*c)*c^4+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d/x^2*( 
c^2*x^2+1)^(1/2)*arcsinh(x*c)*c-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2 
*c^2*x^2-1)/d*x^3*(c^2*x^2+1)*arcsinh(x*c)*c^6-2*b^2*(d*(c^2*x^2+1))^(1/2) 
/(3*c^4*x^4+2*c^2*x^2-1)/d*x^2*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*c^5+1/3*b^ 
2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d/x*c^2+1/3*b^2*(d*(c^2*x^ 
2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d/x^3*arcsinh(x*c)^2-4/3*b^2*(d*(c^2*x 
^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*polylog(2,x*c+(c^2*x^2+1)^(1/2))*c^3-1/3* 
b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d*c^3*(c^2*x^2+1)^(1/2)+ 
2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d*x^5*c^8-1/3*b^2*(d 
*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d*x^3*c^6-2/3*b^2*(d*(c^2*x^2+ 
1))^(1/2)/(3*c^4*x^4+2*c^2*x^2-1)/d*x*c^4-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c 
^2*x^2+1)^(1/2)*(4*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)*x^3*c^3-4*arcsinh(x*c)* 
x^3*c^3-4*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2+2*arcsinh(x*c)*(c^2*x^2+1 
)^(1/2)+x*c)/d/x^3-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*polyl 
og(2,-x*c-(c^2*x^2+1)^(1/2))*c^3+4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1) 
^(1/2)/d*arcsinh(x*c)^2*c^3+a^2*(-1/3/d/x^3*(c^2*d*x^2+d)^(1/2)+2/3*c^2/d/ 
x*(c^2*d*x^2+d)^(1/2))-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2- 
1)/d*x^3*(c^2*x^2+1)*c^6+2*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*c^2*x^2- 
1)/d*x^3*arcsinh(x*c)^2*c^6+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+2*...
 

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(1/2),x, algorithm="frica 
s")
 

Output:

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^ 
2)/(c^2*d*x^6 + d*x^4), x)
 

Sympy [F]

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

integrate((a+b*asinh(c*x))**2/x**4/(c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*asinh(c*x))**2/(x**4*sqrt(d*(c**2*x**2 + 1))), x)
 

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(1/2),x, algorithm="maxim 
a")
 

Output:

-1/3*(4*c^2*log(x)/sqrt(d) + 1/(sqrt(d)*x^2))*a*b*c + 2/3*a*b*(2*sqrt(c^2* 
d*x^2 + d)*c^2/(d*x) - sqrt(c^2*d*x^2 + d)/(d*x^3))*arcsinh(c*x) + 1/3*a^2 
*(2*sqrt(c^2*d*x^2 + d)*c^2/(d*x) - sqrt(c^2*d*x^2 + d)/(d*x^3)) + b^2*int 
egrate(log(c*x + sqrt(c^2*x^2 + 1))^2/(sqrt(c^2*d*x^2 + d)*x^4), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate((b*arcsinh(c*x) + a)^2/(sqrt(c^2*d*x^2 + d)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d\,c^2\,x^2+d}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\frac {2 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, x^{4}}d x \right ) b^{2} x^{3}-2 a^{2} c^{3} x^{3}}{3 \sqrt {d}\, x^{3}} \] Input:

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

Output:

(2*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt(c**2*x**2 + 1)*a**2 + 6*int(a 
sinh(c*x)/(sqrt(c**2*x**2 + 1)*x**4),x)*a*b*x**3 + 3*int(asinh(c*x)**2/(sq 
rt(c**2*x**2 + 1)*x**4),x)*b**2*x**3 - 2*a**2*c**3*x**3)/(3*sqrt(d)*x**3)