3.38 \(\int \frac{x^3 (a+b \cosh ^{-1}(c x))}{(d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=179 \[ \frac{b \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2}-\frac{b \sqrt{c x-1}}{2 c^4 d^2 \sqrt{c x+1}}-\frac{b}{2 c^4 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \cosh ^{-1}(c x)}{2 c^4 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.204298, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5750, 89, 12, 78, 52, 5715, 3716, 2190, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2}-\frac{b \sqrt{c x-1}}{2 c^4 d^2 \sqrt{c x+1}}-\frac{b}{2 c^4 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \cosh ^{-1}(c x)}{2 c^4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5750

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 5715

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && 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{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b \int \frac{x^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac{\int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2 d}\\ &=-\frac{b}{2 c^4 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac{b \int \frac{c^2 x}{\sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{2 c^4 d^2}\\ &=-\frac{b}{2 c^4 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac{b \int \frac{x}{\sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{2 c^2 d^2}\\ &=-\frac{b}{2 c^4 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-1+c x}}{2 c^4 d^2 \sqrt{1+c x}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^4 d^2}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 d^2}+\frac{b \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c^3 d^2}\\ &=-\frac{b}{2 c^4 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-1+c x}}{2 c^4 d^2 \sqrt{1+c x}}+\frac{b \cosh ^{-1}(c x)}{2 c^4 d^2}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^4 d^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}\\ &=-\frac{b}{2 c^4 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-1+c x}}{2 c^4 d^2 \sqrt{1+c x}}+\frac{b \cosh ^{-1}(c x)}{2 c^4 d^2}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^4 d^2}+\frac{b \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 c^4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.620754, size = 209, normalized size = 1.17 \[ \frac{4 b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+4 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-\frac{2 a}{c^2 x^2-1}+2 a \log \left (1-c^2 x^2\right )-b \sqrt{\frac{c x-1}{c x+1}}+\frac{b \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}-2 b \cosh ^{-1}(c x)^2+\frac{b \cosh ^{-1}(c x)}{1-c x}+\frac{b \cosh ^{-1}(c x)}{c x+1}+4 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+4 b \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 c^4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.192, size = 309, normalized size = 1.7 \begin{align*} -{\frac{a}{4\,{d}^{2}{c}^{4} \left ( cx-1 \right ) }}+{\frac{a\ln \left ( cx-1 \right ) }{2\,{d}^{2}{c}^{4}}}+{\frac{a}{4\,{d}^{2}{c}^{4} \left ( cx+1 \right ) }}+{\frac{a\ln \left ( cx+1 \right ) }{2\,{d}^{2}{c}^{4}}}-{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2\,{d}^{2}{c}^{4}}}-{\frac{bx}{2\,{c}^{3}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{x}^{2}}{2\,{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right )}{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{2}{c}^{4}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{2}{c}^{4}}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c^4*a/d^2/(c*x-1)+1/2/c^4*a/d^2*ln(c*x-1)+1/4/c^4*a/d^2/(c*x+1)+1/2/c^4*a/d^2*ln(c*x+1)-1/2/c^4*b/d^2*arc
cosh(c*x)^2-1/2/c^3*b/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+1/2/c^2*b/d^2/(c^2*x^2-1)*x^2-1/2/c^4*b/d^
2/(c^2*x^2-1)*arccosh(c*x)-1/2/c^4*b/d^2/(c^2*x^2-1)+1/c^4*b/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(
1/2))+1/c^4*b/d^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+1/c^4*b/d^2*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*
(c*x+1)^(1/2))+1/c^4*b/d^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, b{\left (\frac{{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{2} + 2 \,{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) \log \left (c x - 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )^{2} - 4 \,{\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right ) - 1\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 2}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - 8 \, \int \frac{{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right ) - 1}{2 \,{\left (c^{8} d^{2} x^{5} - 2 \, c^{6} d^{2} x^{3} + c^{4} d^{2} x +{\left (c^{7} d^{2} x^{4} - 2 \, c^{5} d^{2} x^{2} + c^{3} d^{2}\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} - \frac{1}{2} \, a{\left (\frac{1}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4} d^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*(((c^2*x^2 - 1)*log(c*x + 1)^2 + 2*(c^2*x^2 - 1)*log(c*x + 1)*log(c*x - 1) + (c^2*x^2 - 1)*log(c*x - 1)
^2 - 4*((c^2*x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log(c*x - 1) - 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) +
2)/(c^6*d^2*x^2 - c^4*d^2) - 8*integrate(1/2*((c^2*x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log(c*x - 1) - 1)/(c^
8*d^2*x^5 - 2*c^6*d^2*x^3 + c^4*d^2*x + (c^7*d^2*x^4 - 2*c^5*d^2*x^2 + c^3*d^2)*e^(1/2*log(c*x + 1) + 1/2*log(
c*x - 1))), x)) - 1/2*a*(1/(c^6*d^2*x^2 - c^4*d^2) - log(c^2*x^2 - 1)/(c^4*d^2))

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \operatorname{arcosh}\left (c x\right ) + a x^{3}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3*arccosh(c*x) + a*x^3)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{3} \operatorname{acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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