\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 (d+c^2 d x^2)} \, dx\) [233]

Optimal result

Integrand size = 26, antiderivative size = 297 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=-\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {14 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \]

[Out]

-1/3*b^2*c^2/d/x-1/3*(a+b*arcsinh(c*x))^2/d/x^3+c^2*(a+b*arcsinh(c*x))^2/d/x+2*c^3*(a+b*arcsinh(c*x))^2*arctan
(c*x+(c^2*x^2+1)^(1/2))/d+14/3*b*c^3*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/d+7/3*b^2*c^3*polylog(2
,-c*x-(c^2*x^2+1)^(1/2))/d-2*I*b*c^3*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d+2*I*b*c^3*(a+b
*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d-7/3*b^2*c^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d+2*I*b^2*c
^3*polylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/d-2*I*b^2*c^3*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2)))/d-1/3*b*c*(a+b*arc
sinh(c*x))*(c^2*x^2+1)^(1/2)/d/x^2

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5809, 5789, 4265, 2611, 2320, 6724, 5816, 4267, 2317, 2438, 30} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\frac {2 c^3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{d}+\frac {14 b c^3 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 d}-\frac {2 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}+\frac {2 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {b c \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 d x^2}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {b^2 c^2}{3 d x} \]

[In]

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

[Out]

-1/3*(b^2*c^2)/(d*x) - (b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*d*x^2) - (a + b*ArcSinh[c*x])^2/(3*d*x^
3) + (c^2*(a + b*ArcSinh[c*x])^2)/(d*x) + (2*c^3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/d + (14*b*c^3*
(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(3*d) + (7*b^2*c^3*PolyLog[2, -E^ArcSinh[c*x]])/(3*d) - ((2*I)*b
*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d + ((2*I)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, I*
E^ArcSinh[c*x]])/d - (7*b^2*c^3*PolyLog[2, E^ArcSinh[c*x]])/(3*d) + ((2*I)*b^2*c^3*PolyLog[3, (-I)*E^ArcSinh[c
*x]])/d - ((2*I)*b^2*c^3*PolyLog[3, I*E^ArcSinh[c*x]])/d

Rule 30

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 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 2438

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 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 4265

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] + (-Dist[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] + Dist[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 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 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[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 5809

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] + (-Dist[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] - Dist[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]

Rule 5816

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

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}{3 d x^3}-c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx+\frac {(2 b c) \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {1+c^2 x^2}} \, dx}{3 d} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+c^4 \int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2} \, dx}{3 d}-\frac {\left (b c^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{3 d}-\frac {\left (2 b c^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {c^3 \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{d}-\frac {\left (b c^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{3 d}-\frac {\left (2 b c^3\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {14 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {\left (2 i b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}+\frac {\left (2 i b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d}+\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}-\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {14 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}-\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {\left (2 b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {14 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {\left (2 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {14 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d}-\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {7 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d}+\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c^3 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(297)=594\).

Time = 7.33 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=-\frac {a^2}{3 d x^3}+\frac {a^2 c^2}{d x}+\frac {a^2 c^3 \arctan (c x)}{d}+\frac {2 a b \left (-\frac {c \sqrt {1+c^2 x^2}}{6 x^2}-\frac {\text {arcsinh}(c x)}{3 x^3}+\frac {1}{6} c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )-c^2 \left (-\frac {\text {arcsinh}(c x)}{x}-c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )\right )-\frac {1}{2} i c^4 \left (-\frac {\text {arcsinh}(c x)^2}{2 c}+\frac {2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{c}+\frac {2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c}\right )+\frac {1}{2} i c^4 \left (-\frac {\text {arcsinh}(c x)^2}{2 c}+\frac {2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )}{c}+\frac {2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c}\right )\right )}{d}+\frac {b^2 c^3 \left (-4 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )+14 \text {arcsinh}(c x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-\frac {1}{2} c x \text {arcsinh}(c x)^2 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(c x)\right )-56 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-24 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+24 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+56 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-56 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-48 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+48 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+56 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-48 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+48 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )-2 \text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-\frac {8 \text {arcsinh}(c x)^2 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(c x)\right )}{c^3 x^3}+4 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-14 \text {arcsinh}(c x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{24 d} \]

[In]

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

[Out]

-1/3*a^2/(d*x^3) + (a^2*c^2)/(d*x) + (a^2*c^3*ArcTan[c*x])/d + (2*a*b*(-1/6*(c*Sqrt[1 + c^2*x^2])/x^2 - ArcSin
h[c*x]/(3*x^3) + (c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/6 - c^2*(-(ArcSinh[c*x]/x) - c*ArcTanh[Sqrt[1 + c^2*x^2]]) -
(I/2)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSin
h[c*x]])/c) + (I/2)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/c + (2*PolyLog[2,
I*E^ArcSinh[c*x]])/c)))/d + (b^2*c^3*(-4*Coth[ArcSinh[c*x]/2] + 14*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] - 2*Arc
Sinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - (c*x*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 56*ArcSinh[c*x]*Log[1 - E^(
-ArcSinh[c*x])] - (24*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[
c*x]] + 56*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 56*PolyLog[2, -E^(-ArcSinh[c*x])] - (48*I)*ArcSinh[c*x]*P
olyLog[2, (-I)/E^ArcSinh[c*x]] + (48*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 56*PolyLog[2, E^(-ArcSinh[
c*x])] - (48*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (48*I)*PolyLog[3, I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sech[Ar
cSinh[c*x]/2]^2 - (8*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2]^4)/(c^3*x^3) + 4*Tanh[ArcSinh[c*x]/2] - 14*ArcSinh[c*
x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d)

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{x^{4} \left (c^{2} d \,x^{2}+d \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((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]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx}{d} \]

[In]

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

[Out]

(Integral(a**2/(c**2*x**6 + x**4), x) + Integral(b**2*asinh(c*x)**2/(c**2*x**6 + x**4), x) + Integral(2*a*b*as
inh(c*x)/(c**2*x**6 + x**4), x))/d

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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