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

Optimal. Leaf size=186 \[ -\frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}+\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{\tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{b}{8 c^3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]

[Out]

b/(12*c^3*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + b/(8*c^3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCo
sh[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (x*(a + b*ArcCosh[c*x]))/(8*c^2*d^3*(1 - c^2*x^2)) - ((a + b*ArcCosh[c
*x])*ArcTanh[E^ArcCosh[c*x]])/(4*c^3*d^3) - (b*PolyLog[2, -E^ArcCosh[c*x]])/(8*c^3*d^3) + (b*PolyLog[2, E^ArcC
osh[c*x]])/(8*c^3*d^3)

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Rubi [A]  time = 0.179358, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {5750, 74, 5689, 5694, 4182, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}+\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{\tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^3 d^3}+\frac{b}{8 c^3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b/(12*c^3*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + b/(8*c^3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCo
sh[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (x*(a + b*ArcCosh[c*x]))/(8*c^2*d^3*(1 - c^2*x^2)) - ((a + b*ArcCosh[c
*x])*ArcTanh[E^ArcCosh[c*x]])/(4*c^3*d^3) - (b*PolyLog[2, -E^ArcCosh[c*x]])/(8*c^3*d^3) + (b*PolyLog[2, E^ArcC
osh[c*x]])/(8*c^3*d^3)

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 74

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

Rule 5689

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

Rule 5694

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(c*d)^(-1), Subst[Int[
(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

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 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^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b \int \frac{x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 c d^3}-\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=\frac{b}{12 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 c d^3}-\frac{\int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 c^2 d^2}\\ &=\frac{b}{12 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b}{8 c^3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^3 d^3}\\ &=\frac{b}{12 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b}{8 c^3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^3 d^3}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^3 d^3}\\ &=\frac{b}{12 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b}{8 c^3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}\\ &=\frac{b}{12 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b}{8 c^3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 c^3 d^3}-\frac{b \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}+\frac{b \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 c^3 d^3}\\ \end{align*}

Mathematica [A]  time = 1.73316, size = 287, normalized size = 1.54 \[ \frac{-6 b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+6 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )+\frac{6 a c x}{c^2 x^2-1}+\frac{12 a c x}{\left (c^2 x^2-1\right )^2}+3 a \log (1-c x)-3 a \log (c x+1)+\frac{b \sqrt{c x-1} (c x+2)}{(c x+1)^{3/2}}-\frac{b (c x-2) \sqrt{c x+1}}{(c x-1)^{3/2}}+\frac{3 b \cosh ^{-1}(c x)}{(c x-1)^2}-\frac{3 b \cosh ^{-1}(c x)}{(c x+1)^2}-3 b \left (\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right )-3 b \left (\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right )-\frac{3}{2} b \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (1-e^{\cosh ^{-1}(c x)}\right )\right )+\frac{3}{2} b \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )}{48 c^3 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-((b*(-2 + c*x)*Sqrt[1 + c*x])/(-1 + c*x)^(3/2)) + (b*Sqrt[-1 + c*x]*(2 + c*x))/(1 + c*x)^(3/2) + (12*a*c*x)/
(-1 + c^2*x^2)^2 + (6*a*c*x)/(-1 + c^2*x^2) + (3*b*ArcCosh[c*x])/(-1 + c*x)^2 - (3*b*ArcCosh[c*x])/(1 + c*x)^2
- 3*b*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) - 3*b*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[
c*x]/(1 + c*x)) - (3*b*ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]))/2 + (3*b*ArcCosh[c*x]*(ArcCosh
[c*x] - 4*Log[1 + E^ArcCosh[c*x]]))/2 + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] - 6*b*PolyLog[2, -E^ArcCosh[c*x]]
+ 6*b*PolyLog[2, E^ArcCosh[c*x]])/(48*c^3*d^3)

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Maple [A]  time = 0.153, size = 380, normalized size = 2. \begin{align*}{\frac{a}{16\,{c}^{3}{d}^{3} \left ( cx-1 \right ) ^{2}}}+{\frac{a}{16\,{c}^{3}{d}^{3} \left ( cx-1 \right ) }}+{\frac{a\ln \left ( cx-1 \right ) }{16\,{c}^{3}{d}^{3}}}-{\frac{a}{16\,{c}^{3}{d}^{3} \left ( cx+1 \right ) ^{2}}}+{\frac{a}{16\,{c}^{3}{d}^{3} \left ( cx+1 \right ) }}-{\frac{a\ln \left ( cx+1 \right ) }{16\,{c}^{3}{d}^{3}}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{3}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{b{x}^{2}}{8\,c{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right )x}{8\,{c}^{2}{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{b}{24\,{c}^{3}{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )}{8\,{c}^{3}{d}^{3}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{8\,{c}^{3}{d}^{3}}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{\rm arccosh} \left (cx\right )}{8\,{c}^{3}{d}^{3}}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b}{8\,{c}^{3}{d}^{3}}{\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^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x)

[Out]

1/16/c^3*a/d^3/(c*x-1)^2+1/16/c^3*a/d^3/(c*x-1)+1/16/c^3*a/d^3*ln(c*x-1)-1/16/c^3*a/d^3/(c*x+1)^2+1/16/c^3*a/d
^3/(c*x+1)-1/16/c^3*a/d^3*ln(c*x+1)+1/8*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arccosh(c*x)*x^3+1/8/c*b/d^3/(c^4*x^4-2*c^
2*x^2+1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+1/8/c^2*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arccosh(c*x)*x-1/24/c^3*b/d^3/(c^
4*x^4-2*c^2*x^2+1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)-1/8/c^3*b/d^3*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)
)-1/8*b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^3/d^3+1/8/c^3*b/d^3*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*
(c*x+1)^(1/2))+1/8*b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^3/d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2048*(6144*c^5*integrate(1/32*x^5*log(c*x - 1)/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), x)
- 16*c^4*(2*(5*c^2*x^3 - 3*x)/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3) + 3*log(c*x + 1)/(c^7*d^3) - 3*log(c*x
- 1)/(c^7*d^3)) - 2048*c^4*integrate(1/32*x^4*log(c*x - 1)/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*
d^3), x) + 6*(c*(2*(5*c^2*x^2 + 3*c*x - 6)/(c^10*d^3*x^3 - c^9*d^3*x^2 - c^8*d^3*x + c^7*d^3) - 5*log(c*x + 1)
/(c^7*d^3) + 5*log(c*x - 1)/(c^7*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1)/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d
^3))*c^3 - 16*c^2*(2*(c^2*x^3 + x)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) - log(c*x + 1)/(c^5*d^3) + log(c*x
- 1)/(c^5*d^3)) + 4096*c^2*integrate(1/32*x^2*log(c*x - 1)/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*
d^3), x) + 3*(c*(2*(3*c^2*x^2 - 3*c*x - 2)/(c^8*d^3*x^3 - c^7*d^3*x^2 - c^6*d^3*x + c^5*d^3) - 3*log(c*x + 1)/
(c^5*d^3) + 3*log(c*x - 1)/(c^5*d^3)) - 16*log(c*x - 1)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3))*c - 32*((c^4*
x^4 - 2*c^2*x^2 + 1)*log(c*x + 1)^2 + 2*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1)*log(c*x - 1) + 4*(2*c^3*x^3 + 2
*c*x - (c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x - 1))*log(c*x + sqrt(c*x + 1
)*sqrt(c*x - 1)))/(c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3) + 2048*integrate(-1/16*(2*c^3*x^3 + 2*c*x - (c^4*x^4
- 2*c^2*x^2 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x - 1))/(c^9*d^3*x^7 - 3*c^7*d^3*x^5 + 3*c^5*
d^3*x^3 - c^3*d^3*x + (c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x)
- 2048*integrate(1/32*log(c*x - 1)/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), x))*b + 1/16*a*(2
*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x + 1)/(c^3*d^3) + log(c*x - 1)/(c^3*d^3))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b x^{2} \operatorname{arcosh}\left (c x\right ) + a x^{2}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*x^2*arccosh(c*x) + a*x^2)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a x^{2}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac{b x^{2} \operatorname{acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a*x**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b*x**2*acosh(c*x)/(c**6*x**6 - 3*
c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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