\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 (d+c^2 d x^2)^{3/2}} \, dx\) [309]

Optimal result

Integrand size = 28, antiderivative size = 452 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {16 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}} \]

[Out]

-1/3*b^2*c^2*(c^2*x^2+1)/d/x/(c^2*d*x^2+d)^(1/2)-1/3*(a+b*arcsinh(c*x))^2/d/x^3/(c^2*d*x^2+d)^(1/2)+4/3*c^2*(a
+b*arcsinh(c*x))^2/d/x/(c^2*d*x^2+d)^(1/2)+8/3*c^4*x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b*a
rcsinh(c*x))*(c^2*x^2+1)^(1/2)/d/x^2/(c^2*d*x^2+d)^(1/2)+8/3*c^3*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/d/(c^2
*d*x^2+d)^(1/2)+20/3*b*c^3*(a+b*arcsinh(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d/(c^2*d*x^
2+d)^(1/2)-16/3*b*c^3*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d/(c^2*d*x^2+d)^(1/
2)-b^2*c^3*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d/(c^2*d*x^2+d)^(1/2)-5/3*b^2*c^3*polylog(2
,(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5809, 5787, 5797, 3799, 2221, 2317, 2438, 5799, 5569, 4267, 270} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {20 b c^3 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {c^2 d x^2+d}}+\frac {8 c^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {c^2 d x^2+d}}-\frac {16 b c^3 \sqrt {c^2 x^2+1} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 c^3 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {5 b^2 c^3 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt {c^2 d x^2+d}} \]

[In]

Int[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(3/2)),x]

[Out]

-1/3*(b^2*c^2*(1 + c^2*x^2))/(d*x*Sqrt[d + c^2*d*x^2]) - (b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*d*x^2
*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(3*d*x^3*Sqrt[d + c^2*d*x^2]) + (4*c^2*(a + b*ArcSinh[c*x])^2)/
(3*d*x*Sqrt[d + c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSinh[c*x])^2)/(3*d*Sqrt[d + c^2*d*x^2]) + (8*c^3*Sqrt[1 + c^2
*x^2]*(a + b*ArcSinh[c*x])^2)/(3*d*Sqrt[d + c^2*d*x^2]) + (20*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Arc
Tanh[E^(2*ArcSinh[c*x])])/(3*d*Sqrt[d + c^2*d*x^2]) - (16*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 +
E^(2*ArcSinh[c*x])])/(3*d*Sqrt[d + c^2*d*x^2]) - (b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/
(d*Sqrt[d + c^2*d*x^2]) - (5*b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(3*d*Sqrt[d + c^2*d*x^2
])

Rule 270

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

Rule 2221

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]
- Dist[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 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 3799

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] + Dist[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 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 5797

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}-\frac {1}{3} \left (4 c^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {1}{3} \left (8 c^4\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2}} \, dx}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (8 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 b c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (8 b c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (16 b c^5 \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (16 b c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (16 b c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (32 b c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (8 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {16 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (16 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {16 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (8 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {16 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-a^2+4 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4-b^2 c^4 x^4-a b c x \sqrt {1+c^2 x^2}-2 a b \text {arcsinh}(c x)+8 a b c^2 x^2 \text {arcsinh}(c x)+16 a b c^4 x^4 \text {arcsinh}(c x)-b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-b^2 \text {arcsinh}(c x)^2+4 b^2 c^2 x^2 \text {arcsinh}(c x)^2+8 b^2 c^4 x^4 \text {arcsinh}(c x)^2-8 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-10 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-6 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-10 a b c^3 x^3 \sqrt {1+c^2 x^2} \log (c x)-3 a b c^3 x^3 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+3 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+5 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 d x^3 \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(3/2)),x]

[Out]

(-a^2 + 4*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 - b^2*c^4*x^4 - a*b*c*x*Sqrt[1 + c^2*x^2] - 2*a*b*ArcSinh[
c*x] + 8*a*b*c^2*x^2*ArcSinh[c*x] + 16*a*b*c^4*x^4*ArcSinh[c*x] - b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b^2
*ArcSinh[c*x]^2 + 4*b^2*c^2*x^2*ArcSinh[c*x]^2 + 8*b^2*c^4*x^4*ArcSinh[c*x]^2 - 8*b^2*c^3*x^3*Sqrt[1 + c^2*x^2
]*ArcSinh[c*x]^2 - 10*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 6*b^2*c^3*x^3*
Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 10*a*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[c*x] - 3*a*
b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + 3*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])
] + 5*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*d*x^3*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2607\) vs. \(2(438)=876\).

Time = 0.37 (sec) , antiderivative size = 2608, normalized size of antiderivative = 5.77

method	result	size
default	\(\text {Expression too large to display}\)	\(2608\)
parts	\(\text {Expression too large to display}\)	\(2608\)

[In]

int((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-128/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^2*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5+40/3*b^2*(
d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5*c^8-7/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)
/d^2*x*c^4+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x*c^2+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^
4*x^4+7*c^2*x^2-1)/d^2/x^3*arcsinh(c*x)^2-10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*polylog(2,-c*x-
(c^2*x^2+1)^(1/2))*c^3+16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*arcsinh(c*x)^2*c^3-1/3*b^2*(d*(c^2
*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*c^3*(c^2*x^2+1)^(1/2)-10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1
/2)/d^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c^3+32/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^7*c^
10-b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)*c^3+a^2*(-1/3/d/x^3/(
c^2*d*x^2+d)^(1/2)-4/3*c^2*(-1/d/x/(c^2*d*x^2+d)^(1/2)-2*c^2/d*x/(c^2*d*x^2+d)^(1/2)))-64/3*a*b*(d*(c^2*x^2+1)
)^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5*(c^2*x^2+1)*c^8-32/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)
/d^2*x^3*(c^2*x^2+1)*c^6+128/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*arcsinh(c*x)*c^6+8/3*
a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x*(c^2*x^2+1)*c^4+16*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^
4+7*c^2*x^2-1)/d^2*x*arcsinh(c*x)*c^4+16/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*(c^2*x^2+1)^(
1/2)*arcsinh(c*x)*c^3-8*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x*arcsinh(c*x)*c^2+1/3*a*b*(d*(c
^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x^2*c*(c^2*x^2+1)^(1/2)-32/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4
+7*c^2*x^2-1)/d^2*x^5*(c^2*x^2+1)*c^8+32*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5*arcsinh(c*x
)*c^8-8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*(c^2*x^2+1)*c^6+64/3*b^2*(d*(c^2*x^2+1))^(
1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*arcsinh(c*x)^2*c^6+8*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^
2*x^3*arcsinh(c*x)*c^6+8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^2*c^5*(c^2*x^2+1)^(1/2)+8*b
^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x*arcsinh(c*x)^2*c^4-8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4
*x^4+7*c^2*x^2-1)/d^2*x*arcsinh(c*x)*c^4+8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*(c^2*x^2+1)
^(1/2)*arcsinh(c*x)^2*c^3-8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*(c^2*x^2+1)^(1/2)*arcsinh(
c*x)*c^3-2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*c^3-4*
b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x*arcsinh(c*x)^2*c^2-10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2
*x^2+1)^(1/2)/d^2*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c^3-10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2
)/d^2*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c^3+64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*
x^7*arcsinh(c*x)*c^10+32/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*arcsinh(c*x)*c^3+64/3*a*b*(d*(c^2*x
^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^7*c^10+32*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5
*c^8+8*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*c^6-8/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+
7*c^2*x^2-1)/d^2*x*c^4-8/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*c^3*(c^2*x^2+1)^(1/2)+2/3*a*b
*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x^3*arcsinh(c*x)-2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1
/2)/d^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*c^3-10/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*ln((c*x+(c^2*
x^2+1)^(1/2))^2-1)*c^3-64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5*(c^2*x^2+1)*arcsinh(c*x)
*c^8-32/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*(c^2*x^2+1)*arcsinh(c*x)*c^6-64/3*b^2*(d*(
c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^2*(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^5+8/3*b^2*(d*(c^2*x^2+1))
^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x*(c^2*x^2+1)*arcsinh(c*x)*c^4+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c
^2*x^2-1)/d^2/x^2*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^4*d^2*x^8 + 2*c^2*d^2*x^6 + d^
2*x^4), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

1/3*(8*c^4*x/(sqrt(c^2*d*x^2 + d)*d) + 4*c^2/(sqrt(c^2*d*x^2 + d)*d*x) - 1/(sqrt(c^2*d*x^2 + d)*d*x^3))*a^2 +
integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/((c^2*d*x^2 + d)^(3/2)*x^4) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/
((c^2*d*x^2 + d)^(3/2)*x^4), x)

Giac [F]

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

[In]

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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