\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx\) [238]

Optimal result

Integrand size = 26, antiderivative size = 199 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {b^2 x^2}{4 c^2 d}-\frac {b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d}+\frac {(a+b \text {arcsinh}(c x))^2}{4 c^4 d}+\frac {x^2 (a+b \text {arcsinh}(c x))^2}{2 c^2 d}+\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^4 d}-\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^4 d}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^4 d}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^4 d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.48 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=-\frac {-12 a^2 c^2 x^2+12 a b c x \sqrt {1+c^2 x^2}-24 a b c^2 x^2 \text {arcsinh}(c x)-24 a b \text {arcsinh}(c x)^2+8 b^2 \text {arcsinh}(c x)^3-3 b^2 \cosh (2 \text {arcsinh}(c x))-6 b^2 \text {arcsinh}(c x)^2 \cosh (2 \text {arcsinh}(c x))+24 b^2 \text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+48 a b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+48 a b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+12 a^2 \log \left (1+c^2 x^2\right )+12 a b \log \left (-c x+\sqrt {1+c^2 x^2}\right )-24 b^2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+48 a b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+48 a b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )+6 b^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))}{24 c^4 d} \] Input:

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

Output:

-1/24*(-12*a^2*c^2*x^2 + 12*a*b*c*x*Sqrt[1 + c^2*x^2] - 24*a*b*c^2*x^2*Arc 
Sinh[c*x] - 24*a*b*ArcSinh[c*x]^2 + 8*b^2*ArcSinh[c*x]^3 - 3*b^2*Cosh[2*Ar 
cSinh[c*x]] - 6*b^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + 24*b^2*ArcSinh[c 
*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] + 48*a*b*ArcSinh[c*x]*Log[1 - I*E^ArcSi 
nh[c*x]] + 48*a*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 12*a^2*Log[1 + 
c^2*x^2] + 12*a*b*Log[-(c*x) + Sqrt[1 + c^2*x^2]] - 24*b^2*ArcSinh[c*x]*Po 
lyLog[2, -E^(-2*ArcSinh[c*x])] + 48*a*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + 
48*a*b*PolyLog[2, I*E^ArcSinh[c*x]] - 12*b^2*PolyLog[3, -E^(-2*ArcSinh[c*x 
])] + 6*b^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])/(c^4*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 199, 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 = {6227, 27, 6212, 3042, 26, 4201, 2620, 3011, 2720, 6227, 15, 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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 6227

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/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*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] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 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 [A] (verified)

Time = 1.45 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.72

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {c^{2} x^{2}}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{3}}{3 d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}}{2 d}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{2 d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{4 d}+\frac {b^{2} x^{2} c^{2}}{4 d}+\frac {b^{2}}{8 d}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {a b \operatorname {arcsinh}\left (x c \right )^{2}}{d}+\frac {a b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{d}-\frac {a b x c \sqrt {c^{2} x^{2}+1}}{2 d}+\frac {a b \,\operatorname {arcsinh}\left (x c \right )}{2 d}-\frac {2 a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}}{c^{4}}\)	\(343\)
default	\(\frac {\frac {a^{2} \left (\frac {c^{2} x^{2}}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{3}}{3 d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}}{2 d}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{2 d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{4 d}+\frac {b^{2} x^{2} c^{2}}{4 d}+\frac {b^{2}}{8 d}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {a b \operatorname {arcsinh}\left (x c \right )^{2}}{d}+\frac {a b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{d}-\frac {a b x c \sqrt {c^{2} x^{2}+1}}{2 d}+\frac {a b \,\operatorname {arcsinh}\left (x c \right )}{2 d}-\frac {2 a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}}{c^{4}}\)	\(343\)
parts	\(\frac {a^{2} \left (\frac {x^{2}}{2 c^{2}}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2 c^{4}}\right )}{d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{3}}{3 d \,c^{4}}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{2}}{2 d \,c^{2}}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x}{2 d \,c^{3}}+\frac {b^{2} x^{2}}{4 c^{2} d}+\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{4 d \,c^{4}}+\frac {b^{2}}{8 d \,c^{4}}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}-\frac {b^{2} \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}+\frac {b^{2} \operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 c^{4} d}+\frac {a b \operatorname {arcsinh}\left (x c \right )^{2}}{d \,c^{4}}+\frac {a b \,\operatorname {arcsinh}\left (x c \right ) x^{2}}{d \,c^{2}}-\frac {a b \sqrt {c^{2} x^{2}+1}\, x}{2 d \,c^{3}}+\frac {a b \,\operatorname {arcsinh}\left (x c \right )}{2 d \,c^{4}}-\frac {2 a b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}-\frac {a b \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d \,c^{4}}\)	\(376\)

Input:

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

Output:

1/c^4*(a^2/d*(1/2*c^2*x^2-1/2*ln(c^2*x^2+1))+1/3/d*b^2*arcsinh(x*c)^3+1/2/ 
d*b^2*arcsinh(x*c)^2*x^2*c^2-1/2/d*b^2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+ 
1/4/d*b^2*arcsinh(x*c)^2+1/4/d*b^2*x^2*c^2+1/8/d*b^2-1/d*b^2*arcsinh(x*c)^ 
2*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-1/d*b^2*arcsinh(x*c)*polylog(2,-(x*c+(c^ 
2*x^2+1)^(1/2))^2)+1/2*b^2*polylog(3,-(x*c+(c^2*x^2+1)^(1/2))^2)/d+a*b/d*a 
rcsinh(x*c)^2+a*b/d*arcsinh(x*c)*x^2*c^2-1/2*a*b/d*x*c*(c^2*x^2+1)^(1/2)+1 
/2*a*b/d*arcsinh(x*c)-2*a*b/d*arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2) 
-a*b/d*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2))
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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