3.78 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^{10}} \, dx\)

Optimal. Leaf size=328 \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{315 d x^5}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 d x^9}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(72*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^6*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^
7*d*Sqrt[d - c^2*d*x^2])/(315*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))
/(9*d*x^9) - (4*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(63*d*x^7) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a +
b*ArcCosh[c*x]))/(315*d*x^5) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.51281, antiderivative size = 401, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 97, 12, 103, 95, 5733, 1251, 893} \[ -\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac{4 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(72*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^6*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^
7*d*Sqrt[d - c^2*d*x^2])/(315*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*
x]))/(21*x^7) - (c^4*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*x^5) - (4*c^6*d*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x]))/(315*x^3) - (8*c^8*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(315*x) - (d*(1 - c*x)*(1 + c
*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*x^9) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*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 97

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

Rule 12

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

Rule 103

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

Rule 95

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

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx &=-\frac{\left (d \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^{10}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac{4 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (35+20 c^2 x^2+8 c^4 x^4\right )}{315 x^9} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac{4 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (35+20 c^2 x^2+8 c^4 x^4\right )}{x^9} \, dx}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac{4 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-c^2 x\right )^2 \left (35+20 c^2 x+8 c^4 x^2\right )}{x^5} \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac{4 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{35}{x^5}-\frac{50 c^2}{x^4}+\frac{3 c^4}{x^3}+\frac{4 c^6}{x^2}+\frac{8 c^8}{x}\right ) \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac{4 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac{8 b c^9 d \sqrt{d-c^2 d x^2} \log (x)}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.312996, size = 154, normalized size = 0.47 \[ -\frac{d \sqrt{d-c^2 d x^2} \left (96 c^2 x^2 (c x-1)^{5/2} \left (2 c^2 x^2+5\right ) (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+840 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (48 c^6 x^6+18 c^4 x^4-200 c^2 x^2-192 c^8 x^8 \log (x)+105\right )\right )}{7560 x^9 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]

[Out]

-(d*Sqrt[d - c^2*d*x^2]*(840*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 96*c^2*x^2*(-1 + c*x)^(5/
2)*(1 + c*x)^(5/2)*(5 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) + b*c*x*(105 - 200*c^2*x^2 + 18*c^4*x^4 + 48*c^6*x^6 -
192*c^8*x^8*Log[x])))/(7560*x^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B]  time = 0.387, size = 4259, normalized size = 13. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x)

[Out]

-1/9*a/d/x^9*(-c^2*d*x^2+d)^(5/2)+104/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-27
30*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11/(c*x+1)/(c*x-1)*arccosh(c*x)*c^20-7700/9*b*(-d*(c^2*x^2-1))^(1
/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^7/(c*x+1)/(c*x-1
)*arccosh(c*x)*c^2-212/15*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+621
0*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c^18+3151/15*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^1
2*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)
*c^16-60632/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-
4725*c^2*x^2+1225)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14+59884/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-9
45*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^12-43
264/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*
x^2+1225)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+113594/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^1
0+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-174520/63*b*(-d*
(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^3
/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+19540/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-
2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4+1104/7*b*(-d*(c^2*x^2-1))^(1
/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^4/(c*x+1)^(1/2)/
(c*x-1)^(1/2)*arccosh(c*x)*c^13-120*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c
^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11+64/3*b*(-d*(c^2*x^2-1
))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^12/(c*x+1)^
(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^21-24*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2
730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^19+24/5*b*(-d*(c^2
*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^8/(c*
x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^17-208/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c
^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^15-40/63*b*
(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)
*x^7*c^16-35/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4
725*c^2*x^2+1225)*x/(c*x+1)/(c*x-1)*c^10-16/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*
x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^19+4*b*(-d*(c^2*x^2-1))^(1
/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^8/(c*x+1)^(1/2)/
(c*x-1)^(1/2)*c^17+280/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210
*c^4*x^4-4725*c^2*x^2+1225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9+4189/180*b*(-d*(c^2*x^2-1))^(1/2)*d/(
840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c*x+1)^(1/2)/(c*x-1)
^(1/2)*c^15-1187/60*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*
x^4-4725*c^2*x^2+1225)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13-829/56*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-9
45*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11+259
15/126*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*
x^2+1225)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7-1285/6*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+1
89*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5+21175/216*b*(-d*(c
^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^6/(
c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-1225/72*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-273
0*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c+1225/9*b*(-d*(c^2*x^2-1))^(1/2)*d/
(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^9/(c*x+1)/(c*x-1)*arcc
osh(c*x)+128/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4
-4725*c^2*x^2+1225)*x^17/(c*x+1)/(c*x-1)*c^26-16/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+1
89*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^15/(c*x+1)/(c*x-1)*c^24-344/189*b*(-d*(c^2*x^2-1))^(
1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13/(c*x+1)/(c*x
-1)*c^22-922/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4
-4725*c^2*x^2+1225)*x^11/(c*x+1)/(c*x-1)*c^20+2906/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10
+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c*x+1)/(c*x-1)*c^18+2069/189*b*(-d*(c^2*x^2-1))
^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^7/(c*x+1)/(c*
x-1)*c^16-4639/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x
^4-4725*c^2*x^2+1225)*x^5/(c*x+1)/(c*x-1)*c^14+455/27*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+
189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3/(c*x+1)/(c*x-1)*c^12-64/3*b*(-d*(c^2*x^2-1))^(1/2
)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13/(c*x+1)/(c*x-1)
*arccosh(c*x)*c^22-4/63*a*c^2/d/x^7*(-c^2*d*x^2+d)^(5/2)-30055/504*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-9
45*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9+8/315*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^9*d-16/315*b*(-
d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^9*d-8/315*a*c^4/d/x^5*(-c^2*d*x^2+d)^(5/2)-218
9/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x
^2+1225)*x^5*c^14+350/27*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210
*c^4*x^4-4725*c^2*x^2+1225)*x^3*c^12-35/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-
2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x*c^10-128/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10
*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^15*c^24-16/45*b*(-d*(c^2*x^2-1))^(1/2)*d/(840
*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13*c^22+1384/945*b*(-d*(c^
2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11*c
^20+2306/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-472
5*c^2*x^2+1225)*x^9*c^18

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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)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.9761, size = 1624, normalized size = 4.95 \begin{align*} \left [-\frac{24 \,{\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 96 \,{\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt{-d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1}{\left (x^{4} - 1\right )} \sqrt{-d} - d}{c^{2} x^{4} - x^{2}}\right ) +{\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} -{\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 24 \,{\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt{-c^{2} d x^{2} + d}}{7560 \,{\left (c^{2} x^{11} - x^{9}\right )}}, \frac{192 \,{\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt{d} \arctan \left (\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1}{\left (x^{2} + 1\right )} \sqrt{d}}{c^{2} d x^{4} -{\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 24 \,{\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} -{\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 24 \,{\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt{-c^{2} d x^{2} + d}}{7560 \,{\left (c^{2} x^{11} - x^{9}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="fricas")

[Out]

[-1/7560*(24*(8*b*c^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*sqrt(-
c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 96*(b*c^11*d*x^11 - b*c^9*d*x^9)*sqrt(-d)*log((c^2*d*x^6 + c^2*d
*x^2 - d*x^4 - sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) + (48*b*c^7*d*x
^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(
-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*
c^2*d*x^2 - 35*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9), 1/7560*(192*(b*c^11*d*x^11 - b*c^9*d*x^9)*sqrt(d)*
arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*sqrt(d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - 24*(8*b*c
^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*sqrt(-c^2*d*x^2 + d)*log(
c*x + sqrt(c^2*x^2 - 1)) - (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^
9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^
8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d*x^2 - 35*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9)]

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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)**(3/2)*(a+b*acosh(c*x))/x**10,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{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)/x^10, x)