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

Optimal. Leaf size=180 \[ -\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{1}{25} b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}-\frac{1}{5} b c d^3 (c x-1)^{3/2} (c x+1)^{3/2}+\frac{11}{5} b c d^3 \sqrt{c x-1} \sqrt{c x+1}+b c d^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) \]

[Out]

(11*b*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/5 - (b*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/5 + (b*c*d^3*(-1 + c*
x)^(5/2)*(1 + c*x)^(5/2))/25 - (d^3*(a + b*ArcCosh[c*x]))/x - 3*c^2*d^3*x*(a + b*ArcCosh[c*x]) + c^4*d^3*x^3*(
a + b*ArcCosh[c*x]) - (c^6*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + b*c*d^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]

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Rubi [A]  time = 0.360977, antiderivative size = 239, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {270, 5731, 12, 1610, 1799, 1620, 63, 205} \[ -\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac{b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*b*c*d^3*(1 - c^2*x^2))/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2)^2)/(5*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2)^3)/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/x - 3*c
^2*d^3*x*(a + b*ArcCosh[c*x]) + c^4*d^3*x^3*(a + b*ArcCosh[c*x]) - (c^6*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (b*c
*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

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

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^3 \left (-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6\right )}{5 x \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} \left (b c d^3\right ) \int \frac{-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt{-1+c^2 x^2}} \, dx}{5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-5-15 c^2 x+5 c^4 x^2-c^6 x^3}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{10 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{11 c^2}{\sqrt{-1+c^2 x}}-\frac{5}{x \sqrt{-1+c^2 x}}+3 c^2 \sqrt{-1+c^2 x}-c^2 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{10 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{11 b c d^3 \left (1-c^2 x^2\right )}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )^2}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^3 \left (1-c^2 x^2\right )^3}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{5} c^6 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b c d^3 \sqrt{-1+c^2 x^2} \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.286109, size = 136, normalized size = 0.76 \[ \frac{1}{25} d^3 \left (-5 a c^6 x^5+25 a c^4 x^3-75 a c^2 x-\frac{25 a}{x}+b c \sqrt{c x-1} \sqrt{c x+1} \left (c^4 x^4-7 c^2 x^2+61\right )-\frac{5 b \left (c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right ) \cosh ^{-1}(c x)}{x}-25 b c \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*((-25*a)/x - 75*a*c^2*x + 25*a*c^4*x^3 - 5*a*c^6*x^5 + b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(61 - 7*c^2*x^2 +
c^4*x^4) - (5*b*(5 + 15*c^2*x^2 - 5*c^4*x^4 + c^6*x^6)*ArcCosh[c*x])/x - 25*b*c*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt
[1 + c*x])]))/25

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Maple [A]  time = 0.017, size = 219, normalized size = 1.2 \begin{align*} -{\frac{{d}^{3}a{c}^{6}{x}^{5}}{5}}+{d}^{3}a{c}^{4}{x}^{3}-3\,{d}^{3}a{c}^{2}x-{\frac{{d}^{3}a}{x}}-{\frac{{d}^{3}b{\rm arccosh} \left (cx\right ){c}^{6}{x}^{5}}{5}}+{d}^{3}b{\rm arccosh} \left (cx\right ){c}^{4}{x}^{3}-3\,{d}^{3}b{\rm arccosh} \left (cx\right ){c}^{2}x-{\frac{b{d}^{3}{\rm arccosh} \left (cx\right )}{x}}+{\frac{{d}^{3}b{c}^{5}{x}^{4}}{25}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{7\,{d}^{3}b{c}^{3}{x}^{2}}{25}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{61\,{d}^{3}bc}{25}\sqrt{cx-1}\sqrt{cx+1}}-{{d}^{3}bc\sqrt{cx-1}\sqrt{cx+1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*d^3*a*c^6*x^5+d^3*a*c^4*x^3-3*d^3*a*c^2*x-d^3*a/x-1/5*d^3*b*arccosh(c*x)*c^6*x^5+d^3*b*arccosh(c*x)*c^4*x
^3-3*d^3*b*arccosh(c*x)*c^2*x-d^3*b*arccosh(c*x)/x+1/25*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^5*x^4-7/25*d^3*b*(
c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^2+61/25*b*c*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)-c*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/
2)/(c^2*x^2-1)^(1/2)*arctan(1/(c^2*x^2-1)^(1/2))

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Maxima [A]  time = 1.79264, size = 315, normalized size = 1.75 \begin{align*} -\frac{1}{5} \, a c^{6} d^{3} x^{5} - \frac{1}{75} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac{1}{3} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b c d^{3} -{\left (c \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arcosh}\left (c x\right )}{x}\right )} b d^{3} - \frac{a d^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a*c^6*d^3*x^5 - 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 +
8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*c^6*d^3 + a*c^4*d^3*x^3 + 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2
+ 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^4*d^3 - 3*a*c^2*d^3*x - 3*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c*d^3 - (c
*arcsin(1/(sqrt(c^2)*abs(x))) + arccosh(c*x)/x)*b*d^3 - a*d^3/x

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Fricas [A]  time = 2.43489, size = 549, normalized size = 3.05 \begin{align*} -\frac{5 \, a c^{6} d^{3} x^{6} - 25 \, a c^{4} d^{3} x^{4} + 75 \, a c^{2} d^{3} x^{2} - 50 \, b c d^{3} x \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) - 5 \,{\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 25 \, a d^{3} + 5 \,{\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} -{\left (b c^{6} - 5 \, b c^{4} + 15 \, b c^{2} + 5 \, b\right )} d^{3} x + 5 \, b d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt{c^{2} x^{2} - 1}}{25 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25*(5*a*c^6*d^3*x^6 - 25*a*c^4*d^3*x^4 + 75*a*c^2*d^3*x^2 - 50*b*c*d^3*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) -
5*(b*c^6 - 5*b*c^4 + 15*b*c^2 + 5*b)*d^3*x*log(-c*x + sqrt(c^2*x^2 - 1)) + 25*a*d^3 + 5*(b*c^6*d^3*x^6 - 5*b*
c^4*d^3*x^4 + 15*b*c^2*d^3*x^2 - (b*c^6 - 5*b*c^4 + 15*b*c^2 + 5*b)*d^3*x + 5*b*d^3)*log(c*x + sqrt(c^2*x^2 -
1)) - (b*c^5*d^3*x^5 - 7*b*c^3*d^3*x^3 + 61*b*c*d^3*x)*sqrt(c^2*x^2 - 1))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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