3.79 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^{12}} \, dx\)

Optimal. Leaf size=409 \[ -\frac{16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 d x^5}-\frac{8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{231 d x^7}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{33 d x^9}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 d x^{11}}-\frac{4 b c^9 d \sqrt{d-c^2 d x^2}}{1155 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d \sqrt{d-c^2 d x^2}}{770 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{1386 x^6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{66 x^8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b c^{11} d \log (x) \sqrt{d-c^2 d x^2}}{1155 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(66*x^8*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(1386*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^7*
d*Sqrt[d - c^2*d*x^2])/(770*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c^9*d*Sqrt[d - c^2*d*x^2])/(1155*x^2*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(11*d*x^11) - (2*c^2*(d - c^2*d*x^2)^
(5/2)*(a + b*ArcCosh[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(231*d*x^7) - (16*
c^6*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(1155*d*x^5) + (16*b*c^11*d*Sqrt[d - c^2*d*x^2]*Log[x])/(1155*
Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.606418, antiderivative size = 480, normalized size of antiderivative = 1.17, 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, 1799, 1620} \[ -\frac{16 c^{10} d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac{2 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac{c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}-\frac{4 b c^9 d \sqrt{d-c^2 d x^2}}{1155 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d \sqrt{d-c^2 d x^2}}{770 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{1386 x^6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{66 x^8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b c^{11} d \log (x) \sqrt{d-c^2 d x^2}}{1155 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(66*x^8*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(1386*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^7*
d*Sqrt[d - c^2*d*x^2])/(770*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c^9*d*Sqrt[d - c^2*d*x^2])/(1155*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]))/(33*x^9) - (c^4*d*Sqrt[d - c^2*d*
x^2]*(a + b*ArcCosh[c*x]))/(231*x^7) - (2*c^6*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(385*x^5) - (8*c^8*d
*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(1155*x^3) - (16*c^10*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(
1155*x) - (d*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(11*x^11) + (16*b*c^11*d*Sqrt[d - c
^2*d*x^2]*Log[x])/(1155*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 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]

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^{12}} \, 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^{12}} \, 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 )}{33 x^9}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac{2 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac{16 c^{10} d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right )}{1155 x^{11}} \, 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 )}{33 x^9}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac{2 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac{16 c^{10} d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right )}{x^{11}} \, dx}{1155 \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 )}{33 x^9}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac{2 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac{16 c^{10} d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\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 (105+70 c^2 x+40 c^4 x^2+16 c^6 x^3\right )}{x^6} \, dx,x,x^2\right )}{2310 \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 )}{33 x^9}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac{2 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac{16 c^{10} d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{105}{x^6}-\frac{140 c^2}{x^5}+\frac{5 c^4}{x^4}+\frac{6 c^6}{x^3}+\frac{8 c^8}{x^2}+\frac{16 c^{10}}{x}\right ) \, dx,x,x^2\right )}{2310 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{66 x^8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{1386 x^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^7 d \sqrt{d-c^2 d x^2}}{770 x^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b c^9 d \sqrt{d-c^2 d x^2}}{1155 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 )}{33 x^9}-\frac{c^4 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac{2 c^6 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac{8 c^8 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac{16 c^{10} d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac{16 b c^{11} d \sqrt{d-c^2 d x^2} \log (x)}{1155 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.41739, size = 170, normalized size = 0.42 \[ -\frac{d \sqrt{d-c^2 d x^2} \left (12 c^2 x^2 (c x-1)^{5/2} \left (8 c^4 x^4+20 c^2 x^2+35\right ) (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+630 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (24 c^8 x^8+9 c^6 x^6+5 c^4 x^4-105 c^2 x^2-96 c^{10} x^{10} \log (x)+63\right )\right )}{6930 x^{11} \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]

[Out]

-(d*Sqrt[d - c^2*d*x^2]*(630*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c*x)^(5/
2)*(1 + c*x)^(5/2)*(35 + 20*c^2*x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]) + b*c*x*(63 - 105*c^2*x^2 + 5*c^4*x^4 +
9*c^6*x^6 + 24*c^8*x^8 - 96*c^10*x^10*Log[x])))/(6930*x^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B]  time = 0.493, size = 5518, normalized size = 13.5 \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^12,x)

[Out]

result too large to display

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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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.07987, size = 1806, normalized size = 4.42 \begin{align*} \text{result too large to display} \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^12,x, algorithm="fricas")

[Out]

[-1/6930*(6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b*c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*
x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 48*(b*c^13*d*x^13 - b*c^11*d*x^11)*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)) + (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c
^5*d*x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 6*(16*a*c^12*d*x^12 - 8*a*c^
10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(
c^2*x^13 - x^11), 1/6930*(96*(b*c^13*d*x^13 - b*c^11*d*x^11)*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)) - 6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*
x^8 - b*c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 -
1)) - (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*
d*x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*(16*a*c^12*d*x^12 - 8*a*c^10*
d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2
*x^13 - x^11)]

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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**12,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^{12}}\,{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^12,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)/x^12, x)