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

Optimal. Leaf size=299 \[ \frac{2 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{c^2 d x^2+d}}-\frac{2 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{c^2 d x^2+d}}+\frac{2 c^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2 \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-\frac{4 b c^3 \sqrt{c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left (c^2 x^2+1\right )}{3 x \sqrt{c^2 d x^2+d}} \]

[Out]

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

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Rubi [A]  time = 0.427373, antiderivative size = 299, 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 = {5747, 5723, 5659, 3716, 2190, 2279, 2391, 5661, 264} \[ -\frac{2 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt{c^2 d x^2+d}}+\frac{2 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt{c^2 d x^2+d}}+\frac{2 c^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2 \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3}-\frac{4 b c^3 \sqrt{c^2 x^2+1} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left (c^2 x^2+1\right )}{3 x \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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

Mathematica [A]  time = 0.629257, size = 278, normalized size = 0.93 \[ \frac{2 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+2 a^2 c^4 x^4+a^2 c^2 x^2-a^2-a b c x \sqrt{c^2 x^2+1}-4 a b c^3 x^3 \sqrt{c^2 x^2+1} \log (c x)-b \sinh ^{-1}(c x) \left (-2 a \left (2 c^4 x^4+c^2 x^2-1\right )+b c x \sqrt{c^2 x^2+1}+4 b c^3 x^3 \sqrt{c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-b^2 c^4 x^4-b^2 c^2 x^2+b^2 \left (2 c^4 x^4-2 c^3 x^3 \sqrt{c^2 x^2+1}+c^2 x^2-1\right ) \sinh ^{-1}(c x)^2}{3 x^3 \sqrt{c^2 d x^2+d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.309, size = 2147, normalized size = 7.2 \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)^(1/2),x)

[Out]

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

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

[Out]

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

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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}}{\sqrt{c^{2} d x^{2} + d} 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)^(1/2),x, algorithm="giac")

[Out]

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