\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^2} \, dx\) [11]

Optimal result

Integrand size = 23, antiderivative size = 210 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {b (a+b \text {arcsinh}(c x))}{c d^2 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c d^2}-\frac {b^2 \arctan (c x)}{c d^2}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c d^2}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c d^2}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c d^2}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c d^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\frac {a^2 x}{1+c^2 x^2}+\frac {a^2 \arctan (c x)}{c}+\frac {2 a b \left (\sqrt {1+c^2 x^2}+c x \text {arcsinh}(c x)+i \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+i c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-i \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-i c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-i \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{c+c^3 x^2}+\frac {2 b^2 \left (\frac {\text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {c x \text {arcsinh}(c x)^2}{2+2 c^2 x^2}-\frac {1}{2} i \left (-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )\right )}{c}}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6203, 27, 6204, 3042, 4668, 3011, 2720, 6213, 216, 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.

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(x*(a + b*ArcSinh[c*x])^2)/(2*d^2*(1 + c^2*x^2)) - (b*c*(-((a + b*ArcSinh[ 
c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[c*x])/c^2))/d^2 + (2*(a + b*Arc 
Sinh[c*x])^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-((a + b*ArcSinh[c*x])*Poly 
Log[2, (-I)*E^ArcSinh[c*x]]) + b*PolyLog[3, (-I)*E^ArcSinh[c*x]]) - (2*I)* 
b*(-((a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]]) + b*PolyLog[3, I*E 
^ArcSinh[c*x]]))/(2*c*d^2)
 

Defintions of rubi rules used

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 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 4668

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

rule 6203

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + 
c^2*x^2)^p]   Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 6204

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

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{\left (c^{2} d \,x^{2}+d \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*a^2*(x/(c^2*d^2*x^2 + d^2) + arctan(c*x)/(c*d^2)) + integrate(b^2*log( 
c*x + sqrt(c^2*x^2 + 1))^2/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2) + 2*a*b*log 
(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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