3.92 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))}{x^6} \, dx\)

Optimal. Leaf size=293 \[ \frac{c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac{11 b c^3 d^2 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{23 b c^5 d^2 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (11*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(30*
x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x + (c^2*d*(d - c^2*d*x
^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*x^5) + (c^5*d^2*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (23*b*c^5*d^2*Sqrt[d - c^2*d*x^2]
*Log[x])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.947257, antiderivative size = 324, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 5740, 5738, 29, 5676, 14, 266, 43} \[ \frac{c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac{d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac{11 b c^3 d^2 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{23 b c^5 d^2 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (11*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(30*
x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x + (c^2*d^2*(1 - c*x)*
(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*x^3) - (d^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]
*(a + b*ArcCosh[c*x]))/(5*x^5) + (c^5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]) + (23*b*c^5*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5798

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

Rule 5740

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x]
+ (-Dist[(2*e1*e2*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c
*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 1)*Sqrt[1 + c*x]*Sq
rt[-1 + c*x]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b
, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
&& IntegerQ[p - 1/2]

Rule 5738

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 1)), x
] + (-Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)
*(a + b*ArcCosh[c*x])^(n - 1), x], x] - Dist[(c^2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f^2*(m + 1)*Sqrt[1 + c*x]*
Sqrt[-1 + c*x]), Int[((f*x)^(m + 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]) /; FreeQ[{
a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^6} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^6} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-1+c^2 x^2\right )^2}{x^5} \, dx}{5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (-1+c^2 x\right )^2}{x^3} \, dx,x,x^2\right )}{10 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{-1+c^2 x^2}{x^3} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{2 c^2}{x^2}+\frac{c^4}{x}\right ) \, dx,x,x^2\right )}{10 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{1}{x^3}+\frac{c^2}{x}\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^6 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{11 b c^3 d^2 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac{c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac{d^2 (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac{c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{23 b c^5 d^2 \sqrt{d-c^2 d x^2} \log (x)}{15 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 3.18745, size = 400, normalized size = 1.37 \[ \frac{d^2 \left (8 a d \sqrt{\frac{c x-1}{c x+1}} \left (c^2 x^2-1\right ) \left (23 c^4 x^4-11 c^2 x^2+3\right )+120 a c^5 \sqrt{d} x^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-60 b c^4 d x^4 (1-c x) \left (2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (2 \log (c x)+\cosh ^{-1}(c x)^2\right )\right )+40 b c^2 d x^2 (1-c x) \left (2 c^3 x^3 \log (c x)+c x-2 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)\right )-b d (1-c x) \left (c x (10 \log (c x)+3)+20 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)+\log (c x) \cosh \left (5 \cosh ^{-1}(c x)\right )+(5 \log (c x)-1) \cosh \left (3 \cosh ^{-1}(c x)\right )-5 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )\right )}{120 x^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

(d^2*(8*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + c^2*x^2)*(3 - 11*c^2*x^2 + 23*c^4*x^4) + 120*a*c^5*Sqrt[d]*x^5*Sq
rt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 40*b
*c^2*d*x^2*(1 - c*x)*(c*x - 2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + 2*c^3*x^3*Log[c*x]) - 60
*b*c^4*d*x^4*(1 - c*x)*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x]
)) - b*d*(1 - c*x)*(20*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + Cosh[5*ArcCosh[c*x]]*Log[c*x] + Cos
h[3*ArcCosh[c*x]]*(-1 + 5*Log[c*x]) + c*x*(3 + 10*Log[c*x]) - 5*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - ArcCosh[c*
x]*Sinh[5*ArcCosh[c*x]])))/(120*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.275, size = 2429, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x)

[Out]

-8/15*a*c^4/d/x*(-c^2*d*x^2+d)^(7/2)-1173*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-7
5*c^2*x^2+9)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11+1495/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x
^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9-115*b*(-d*(c^2*x^2-1
))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*
c^7-1587*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^9/(c*x+1)/(c*x-1)*
arccosh(c*x)*c^14+3519*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^7/(c
*x+1)/(c*x-1)*arccosh(c*x)*c^12-9595/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c
^2*x^2+9)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+5318/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+
325*c^4*x^4-75*c^2*x^2+9)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-9602/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*
x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+777/5*b*(-d*(c^2*x^2-1))^(1/2)*d^
2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-117/5*b*(-d*(c^2*x^2-
1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-1/5*a/d
/x^5*(-c^2*d*x^2+d)^(7/2)-8/15*a*c^6*x*(-c^2*d*x^2+d)^(5/2)-9/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-76
5*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c+759/2*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035
*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11-1329/4*b*(-d*(c^2*x^2-1))^
(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9+1889/12*b*(-
d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c
^7+69/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/(c*x+1)^(1/2)/(c*x-1)
^(1/2)*arccosh(c*x)*c^5+2/15*a*c^2/d/x^3*(-c^2*d*x^2+d)^(7/2)+207/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8
-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^3/(c*x+1)/(c*x-1)*c^8-69/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^
8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x/(c*x+1)/(c*x-1)*c^6+9/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-7
65*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)-5819/30*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(10
35*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^9/(c*x+1)/(c*x-1)*c^14+18791/60*b*(-d*(c^2*x^2-1))^(1/2)*d^
2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^7/(c*x+1)/(c*x-1)*c^12-943/6*b*(-d*(c^2*x^2-1))^(1/2)*
d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^5/(c*x+1)/(c*x-1)*c^10+141/20*b*(-d*(c^2*x^2-1))^(1/
2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-7153/60*b*(-d*(
c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^5*c^10+759/20*b*(-d*(c^2*x^2-1))^(
1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^3*c^8-69/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*
c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x*c^6+5819/30*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6
*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^7*c^12-2/3*a*c^6*d*x*(-c^2*d*x^2+d)^(3/2)-a*c^6*d^2*x*(-c^2*d*x^2+d)^(1/2)-a*
c^6*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1587*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8
*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13-46/15*b*(-d*(c^2*
x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^5*d^2+1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x
+1)^(1/2)*arccosh(c*x)^2*c^5*d^2+23/15*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))^2+1)*c^5*d^2-175/4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^
2*x^2+9)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5

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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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,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{{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**6,x)

[Out]

Timed out

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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 )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)/x^6, x)