\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^2} \, dx\) [247]

Optimal result

Integrand size = 26, antiderivative size = 213 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {b x (a+b \text {arcsinh}(c x))}{c^3 d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^4 d^2}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^4 d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^4 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^4 d^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.50 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\frac {a^2}{1+c^2 x^2}-\frac {a b \left (\sqrt {1+c^2 x^2}-i \text {arcsinh}(c x)\right )}{i+c x}-\frac {a b \left (\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)\right )}{-i+c x}-a b \text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-a b \text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )+a^2 \log \left (1+c^2 x^2\right )+4 a b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+4 a b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )+2 b^2 \left (-\frac {c x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\text {arcsinh}(c x)^2}{2+2 c^2 x^2}+\frac {1}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \log \left (1+c^2 x^2\right )-\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{2 c^4 d^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.70 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6225, 27, 6212, 3042, 26, 4201, 2620, 3011, 2720, 6225, 240, 6198, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6212

\(\displaystyle \frac {b \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c d^2}+\frac {\int \frac {c x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i (a+b \text {arcsinh}(c x))^2 \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (a+b \text {arcsinh}(c x))^2 \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{c d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 27

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

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, 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 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 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 6198

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

rule 6212

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

rule 6225

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

Time = 1.29 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {1}{2 c^{2} x^{2}+2}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\frac {\left (2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )+2\right ) \operatorname {arcsinh}\left (x c \right )}{2 c^{2} x^{2}+2}+\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}}{c^{4}}\)	\(320\)
default	\(\frac {\frac {a^{2} \left (\frac {1}{2 c^{2} x^{2}+2}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\frac {\left (2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )+2\right ) \operatorname {arcsinh}\left (x c \right )}{2 c^{2} x^{2}+2}+\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}}{c^{4}}\)	\(320\)
parts	\(\frac {a^{2} \left (\frac {1}{2 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2 c^{4}}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\frac {\left (2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )+2\right ) \operatorname {arcsinh}\left (x c \right )}{2 c^{2} x^{2}+2}+\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2} c^{4}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2} c^{4}}\)	\(328\)

Input:

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

Output:

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

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

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

Output:

integral((b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^3)/(c^4* 
d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

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

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

Output:

(Integral(a**2*x**3/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(b**2*x**3 
*asinh(c*x)**2/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(2*a*b*x**3*asi 
nh(c*x)/(c**4*x**4 + 2*c**2*x**2 + 1), x))/d**2
 

Maxima [F]

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

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

Output:

1/2*a^2*(1/(c^6*d^2*x^2 + c^4*d^2) + log(c^2*x^2 + 1)/(c^4*d^2)) + 1/2*(b^ 
2 + (b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))^2/( 
c^6*d^2*x^2 + c^4*d^2) - integrate(-(2*a*b*c^4*x^4 - b^2*c^2*x^2 - b^2 - ( 
b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1) + (2*a*b*c^3*x^3 - b^2 
*c*x - (b^2*c^3*x^3 + b^2*c*x)*log(c^2*x^2 + 1))*sqrt(c^2*x^2 + 1))*log(c* 
x + sqrt(c^2*x^2 + 1))/(c^8*d^2*x^5 + 2*c^6*d^2*x^3 + c^4*d^2*x + (c^7*d^2 
*x^4 + 2*c^5*d^2*x^2 + c^3*d^2)*sqrt(c^2*x^2 + 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*int((asinh(c*x)*x**3)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*a*b*c**6*x**2 + 
4*int((asinh(c*x)*x**3)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*a*b*c**4 + 2*int( 
(asinh(c*x)**2*x**3)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*b**2*c**6*x**2 + 2*i 
nt((asinh(c*x)**2*x**3)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*b**2*c**4 + log(c 
**2*x**2 + 1)*a**2*c**2*x**2 + log(c**2*x**2 + 1)*a**2 - a**2*c**2*x**2)/( 
2*c**4*d**2*(c**2*x**2 + 1))