\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x (d+c^2 d x^2)^2} \, dx\) [239]

Optimal result

Integrand size = 26, antiderivative size = 193 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5811, 5799, 5569, 4267, 2611, 2320, 6724, 5787, 266} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{d^2}-\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {c^2 x^2+1}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}+\frac {b^2 \log \left (c^2 x^2+1\right )}{2 d^2} \]

[In]

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

[Out]

-((b*c*x*(a + b*ArcSinh[c*x]))/(d^2*Sqrt[1 + c^2*x^2])) + (a + b*ArcSinh[c*x])^2/(2*d^2*(1 + c^2*x^2)) - (2*(a
+ b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d^2 + (b^2*Log[1 + c^2*x^2])/(2*d^2) - (b*(a + b*ArcSinh[c*x
])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d^2 + (b*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2 + (b^2*P
olyLog[3, -E^(2*ArcSinh[c*x])])/(2*d^2) - (b^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^2)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[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 5787

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

Rule 5799

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

Rule 5811

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {a+b \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )} \, dx}{d} \\ & = -\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}+\frac {\left (b^2 c^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{d^2} \\ & = -\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 \text {Subst}\left (\int (a+b x)^2 \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x)\right )}{d^2} \\ & = -\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}+\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2} \\ & = -\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}-\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2} \\ & = -\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \\ & = -\frac {b c x (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.65 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.22 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^2} \, dx=\frac {\frac {a^2}{1+c^2 x^2}-\frac {a b \left (\sqrt {1+c^2 x^2}-i \text {arcsinh}(c x)\right )}{i+c x}-\frac {a b \left (\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)\right )}{-i+c x}-2 a b \text {arcsinh}(c x)^2+4 a b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+2 a^2 \log (c x)-a^2 \log \left (1+c^2 x^2\right )+a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )+a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+2 a b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+2 b^2 \left (\frac {i \pi ^3}{24}-\frac {c x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\text {arcsinh}(c x)^2}{2+2 c^2 x^2}-\frac {2}{3} \text {arcsinh}(c x)^3-\text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \log \left (1+c^2 x^2\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )}{2 d^2} \]

[In]

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

[Out]

(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*ArcSin
h[c*x]))/(-I + c*x) - 2*a*b*ArcSinh[c*x]^2 + 4*a*b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*a^2*Log[c*x] -
a^2*Log[1 + c^2*x^2] + a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^A
rcSinh[c*x]]) + a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]
]) + 2*a*b*PolyLog[2, E^(2*ArcSinh[c*x])] + 2*b^2*((I/24)*Pi^3 - (c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ArcSin
h[c*x]^2/(2 + 2*c^2*x^2) - (2*ArcSinh[c*x]^3)/3 - ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] + 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])] + ArcSinh[c*
x]*PolyLog[2, E^(2*ArcSinh[c*x])] + PolyLog[3, -E^(-2*ArcSinh[c*x])]/2 - PolyLog[3, E^(2*ArcSinh[c*x])]/2))/(2
*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(531\) vs. \(2(228)=456\).

Time = 0.26 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.76

method	result	size
derivativedivides	\(\frac {a^{2} \left (\ln \left (c x \right )+\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 {\left (2 c^{2} x^{2}-2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )+2\right ) \operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}+2}-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (\frac {-c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(532\)
default	\(\frac {a^{2} \left (\ln \left (c x \right )+\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 {\left (2 c^{2} x^{2}-2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )+2\right ) \operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}+2}-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (\frac {-c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(532\)
parts	\(\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {a^{2} \ln \left (x \right )}{d^{2}}+\frac {b^{2} \left (\frac {\left (2 c^{2} x^{2}-2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )+2\right ) \operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}+2}-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (\frac {-c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(541\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a**2/(c**4*x**5 + 2*c**2*x**3 + x), x) + Integral(b**2*asinh(c*x)**2/(c**4*x**5 + 2*c**2*x**3 + x),
x) + Integral(2*a*b*asinh(c*x)/(c**4*x**5 + 2*c**2*x**3 + x), x))/d**2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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