\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 (d+c^2 d x^2)^3} \, dx\) [53]

Optimal result

Integrand size = 24, antiderivative size = 232 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \]

[Out]

-1/2*b*c/d^3/x/(c^2*x^2+1)^(3/2)-5/12*b*c^3*x/d^3/(c^2*x^2+1)^(3/2)-3/4*c^2*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)
^2+1/2*(-a-b*arcsinh(c*x))/d^3/x^2/(c^2*x^2+1)^2-3/2*c^2*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)+6*c^2*(a+b*arcsinh
(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^3+3/2*b*c^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^3-3/2*b*c^2*po
lylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)/d^3+2/3*b*c^3*x/d^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5809, 5811, 5799, 5569, 4267, 2317, 2438, 197, 198, 277} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {6 c^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^3}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (c^2 x^2+1\right )}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {b c}{2 d^3 x \left (c^2 x^2+1\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {c^2 x^2+1}}-\frac {5 b c^3 x}{12 d^3 \left (c^2 x^2+1\right )^{3/2}} \]

[In]

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

[Out]

-1/2*(b*c)/(d^3*x*(1 + c^2*x^2)^(3/2)) - (5*b*c^3*x)/(12*d^3*(1 + c^2*x^2)^(3/2)) + (2*b*c^3*x)/(3*d^3*Sqrt[1
+ c^2*x^2]) - (3*c^2*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2) - (a + b*ArcSinh[c*x])/(2*d^3*x^2*(1 + c^2*
x^2)^2) - (3*c^2*(a + b*ArcSinh[c*x]))/(2*d^3*(1 + c^2*x^2)) + (6*c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSin
h[c*x])])/d^3 + (3*b*c^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d^3) - (3*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])])/(
2*d^3)

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && 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 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 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 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 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 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])

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\left (3 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )^{5/2}} \, dx}{2 d^3} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}+\frac {\left (3 b c^3\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}-\frac {\left (2 b c^3\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{d^3}-\frac {\left (3 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx}{d} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}+\frac {\left (b c^3\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^3}-\frac {\left (4 b c^3\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^3}+\frac {\left (3 b c^3\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^3}-\frac {\left (3 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d^2} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}-\frac {\left (3 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d^3} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}-\frac {\left (6 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{d^3} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^3} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {\left (3 b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \\ & = -\frac {b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 b c^3 x}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))}{2 d^3 \left (1+c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {-\frac {18 b c \sqrt {1+c^2 x^2}}{x}+\frac {9 b c \left (1+2 c^2 x^2\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c \left (3+12 c^2 x^2+8 c^4 x^4\right )}{x \left (1+c^2 x^2\right )^{3/2}}-18 b c^2 \text {arcsinh}(c x)^2-\frac {18 (a+b \text {arcsinh}(c x))}{x^2}+\frac {3 (a+b \text {arcsinh}(c x))}{\left (x+c^2 x^3\right )^2}+\frac {9 (a+b \text {arcsinh}(c x))}{x^2+c^2 x^4}+\frac {18 c^2 (a+b \text {arcsinh}(c x))^2}{b}+36 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+36 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+18 a c^2 \log \left (1+c^2 x^2\right )+36 b c^2 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+36 b c^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-18 c^2 \left (2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{12 d^3} \]

[In]

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

[Out]

((-18*b*c*Sqrt[1 + c^2*x^2])/x + (9*b*c*(1 + 2*c^2*x^2))/(x*Sqrt[1 + c^2*x^2]) + (b*c*(3 + 12*c^2*x^2 + 8*c^4*
x^4))/(x*(1 + c^2*x^2)^(3/2)) - 18*b*c^2*ArcSinh[c*x]^2 - (18*(a + b*ArcSinh[c*x]))/x^2 + (3*(a + b*ArcSinh[c*
x]))/(x + c^2*x^3)^2 + (9*(a + b*ArcSinh[c*x]))/(x^2 + c^2*x^4) + (18*c^2*(a + b*ArcSinh[c*x])^2)/b + 36*b*c^2
*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 36*b*c^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x
])/c] + 18*a*c^2*Log[1 + c^2*x^2] + 36*b*c^2*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 36*b*c^2*PolyLog[2, (
Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 18*c^2*(2*(a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])] + b*PolyLog[2, E^(2
*ArcSinh[c*x])]))/(12*d^3)

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{c^{2} x^{2}+1}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{3}}+\frac {b \left (-\frac {-8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 c x \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (c x \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\right )\)	\(344\)
default	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{c^{2} x^{2}+1}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{3}}+\frac {b \left (-\frac {-8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 c x \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (c x \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\right )\)	\(344\)
parts	\(\frac {a \left (\frac {c^{4} \left (-\frac {2}{c^{2} \left (c^{2} x^{2}+1\right )}-\frac {1}{2 c^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{c^{2}}\right )}{2}-\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )\right )}{d^{3}}+\frac {b \,c^{2} \left (-\frac {-8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 c x \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (c x \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\)	\(356\)

[In]

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

[Out]

c^2*(a/d^3*(-1/2/c^2/x^2-3*ln(c*x)-1/4/(c^2*x^2+1)^2-1/(c^2*x^2+1)+3/2*ln(c^2*x^2+1))+b/d^3*(-1/12/(c^4*x^4+2*
c^2*x^2+1)/c^2/x^2*(-8*c^5*x^5*(c^2*x^2+1)^(1/2)+8*c^6*x^6+18*arcsinh(c*x)*c^4*x^4-3*c^3*x^3*(c^2*x^2+1)^(1/2)
+16*c^4*x^4+27*arcsinh(c*x)*c^2*x^2+6*c*x*(c^2*x^2+1)^(1/2)+8*c^2*x^2+6*arcsinh(c*x))-3*arcsinh(c*x)*ln(1+c*x+
(c^2*x^2+1)^(1/2))-3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+3*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+3/2*poly
log(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-3*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-3*polylog(2,c*x+(c^2*x^2+1)^(1/2)
)))

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsinh(c*x) + a)/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^3), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx}{d^{3}} \]

[In]

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

[Out]

(Integral(a/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3), x) + Integral(b*asinh(c*x)/(c**6*x**9 + 3*c**4*x**
7 + 3*c**2*x**5 + x**3), x))/d**3

Maxima [F]

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

[In]

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

[Out]

-1/4*a*((6*c^4*x^4 + 9*c^2*x^2 + 2)/(c^4*d^3*x^6 + 2*c^2*d^3*x^4 + d^3*x^2) - 6*c^2*log(c^2*x^2 + 1)/d^3 + 12*
c^2*log(x)/d^3) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*
x^3), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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