3.309 \(\int \frac{(a+b \sinh ^{-1}(c x))^2}{x^4 (d+c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=452 \[ -\frac{b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{c^2 d x^2+d}}-\frac{16 b c^3 \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{20 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt{c^2 d x^2+d}} \]

[Out]

-(b^2*c^2*(1 + c^2*x^2))/(3*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*S
qrt[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])*ArcTa
nh[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])

________________________________________________________________________________________

Rubi [A]  time = 0.840918, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5747, 5687, 5714, 3718, 2190, 2279, 2391, 5720, 5461, 4182, 264} \[ -\frac{b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{c^2 d x^2+d}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{c^2 d x^2+d}}-\frac{16 b c^3 \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}+\frac{20 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*c^2*(1 + c^2*x^2))/(3*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*S
qrt[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])*ArcTa
nh[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 5747

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*d^IntPart[p]*(d + e*x^
2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSin
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1] && Int
egerQ[m]

Rule 5687

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*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
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 5714

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 3718

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 5720

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 5461

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 4182

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 264

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}-\frac{1}{3} \left (4 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{1}{3} \left (8 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{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 \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{16 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{16 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}+\frac{5 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d+c^2 d x^2}}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt{d+c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{8 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt{d+c^2 d x^2}}+\frac{20 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{16 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}-\frac{b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt{d+c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.848985, size = 438, normalized size = 0.97 \[ \frac{3 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+8 a^2 c^4 x^4+4 a^2 c^2 x^2-a^2-a b c x \sqrt{c^2 x^2+1}-10 a b c^3 x^3 \sqrt{c^2 x^2+1} \log (c x)-3 a b c^3 x^3 \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )+16 a b c^4 x^4 \sinh ^{-1}(c x)+8 a b c^2 x^2 \sinh ^{-1}(c x)-2 a b \sinh ^{-1}(c x)-b^2 c^4 x^4-b^2 c^2 x^2+8 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-8 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2+4 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-10 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )-b^2 \sinh ^{-1}(c x)^2}{3 d x^3 \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[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])

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Maple [B]  time = 0.314, size = 2609, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+4/3*a^2*c^2/d/x/(c^2*d*x^2+d)^(1/
2)+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*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*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)-
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*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*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*arcsinh(c*x)*(c^2*x^2+1)*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*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c-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*arcsinh(c*x)*(c^2*x^2+1)*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*a
rcsinh(c*x)*(c^2*x^2+1)*c^6-64/3*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^4-128/3*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*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^
5-32/3*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+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^6-8/3*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+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)+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^3-1/3*a^2/d/x^3/(c^2*d*x^2+d)^(1/2)+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-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^3-10/3*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^3-10/
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^3-1/3*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)+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+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+8/3*a^2*c
^4/d*x/(c^2*d*x^2+d)^(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^4-4*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+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*arcsinh(c*x)^2*(c^2*x^2+1)
^(1/2)*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*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*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^3-10/3*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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{8} + 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)