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

Optimal. Leaf size=294 \[ -\frac{b^2 c^3 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{c^2 x^2+1}}+\frac{c^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{2 b c^3 \sqrt{c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d}}{3 x}+\frac{b^2 c^3 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt{c^2 x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.284515, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {5723, 5728, 277, 215, 5659, 3716, 2190, 2279, 2391} \[ \frac{b^2 c^3 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{c^2 x^2+1}}-\frac{c^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}+\frac{2 b c^3 \sqrt{c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d}}{3 x}+\frac{b^2 c^3 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 5723

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[(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*Arc
Sinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 5728

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((f*x
)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1
)*(1 + c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*A
rcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rule 277

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 5659

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && 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]

Rubi steps

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

Mathematica [A]  time = 0.778704, size = 240, normalized size = 0.82 \[ -\frac{\sqrt{c^2 d x^2+d} \left (b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+a^2 c^2 x^2 \sqrt{c^2 x^2+1}+a^2 \sqrt{c^2 x^2+1}-2 a b c^3 x^3 \log (c x)-b \sinh ^{-1}(c x) \left (-2 a \left (c^2 x^2+1\right )^{3/2}+2 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-b c x\right )+a b c x+b^2 c^2 x^2 \sqrt{c^2 x^2+1}+b^2 \left (-c^3 x^3+c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}\right ) \sinh ^{-1}(c x)^2\right )}{3 x^3 \sqrt{c^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.298, size = 2557, normalized size = 8.7 \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*(c^2*d*x^2+d)^(1/2)/x^4,x)

[Out]

-2*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8-6*a*b*(d*(c^2*x^2+1))^(1
/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6-20/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x
^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4-10/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x/(c^2*x^2+1)*arcs
inh(c*x)*c^2+2*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^7+2*a*b*
(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5+2/3*b^2*(d*(c^2*x^2+1))^(
1/2)/(c^2*x^2+1)^(1/2)*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)/(3*c^4*x^4+3*c^2*x^
2+1)/x/(c^2*x^2+1)*c^2+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7+b^2*(d*(c^2
*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^
2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1
/2)*arcsinh(c*x)*c^3-1/3*a^2/d/x^3*(c^2*d*x^2+d)^(3/2)+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x
*arcsinh(c*x)*c^4+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*arcsinh(c*x)*c^6+1/3*a*b*(d*(c^2*x
^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*c^6+1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x*c^4-4/3*a
*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3+2/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*ln
((c*x+(c^2*x^2+1)^(1/2))^2-1)*c^3-a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*c^3-1/3*
b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x)^2+2/3*b^2*(d*(c^2*x^2+1))^(1/2)
/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c^3+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2
)*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c^3-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*
x^2+1)*c^8-5/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-4/3*b^2*(d*(c^2*x^2+1))^(
1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^2/(c^2*
x^2+1)^(1/2)*arcsinh(c*x)*c+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*
x)^2*c^7+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^5-b^2*(d*(c^
2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5-a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^
4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+2/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)
^(1/2)*arcsinh(c*x)*c^3-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*c-1/3*a*b*
(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-2/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c
^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-2/3*a*b*
(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x)-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^
4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)^2*c^8-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c
^2*x^2+1)*arcsinh(c*x)*c^8-3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)^2*
c^6-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6-10/3*b^2*(d*(c^2*x^
2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)^2*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+
3*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4-5/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x/(c^2*x^2+1
)*arcsinh(c*x)^2*c^2+2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c^3-2/3*
b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*
x^2+1)/(c^2*x^2+1)^(1/2)*c^3-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*c^6

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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*(c^2*d*x^2+d)^(1/2)/x^4,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 )}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/x^4,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)/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{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*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="giac")

[Out]

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