3.95 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))}{x^{12}} \, dx\)

Optimal. Leaf size=385 \[ -\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{693 d x^7}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {c x-1} \sqrt {c x+1}}-\frac {8 b c^{11} d^2 \log (x) \sqrt {d-c^2 d x^2}}{693 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2}}{693 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{924 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {113 b c^5 d^2 \sqrt {d-c^2 d x^2}}{4158 x^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {23 b c^3 d^2 \sqrt {d-c^2 d x^2}}{792 x^8 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

-1/11*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/d/x^11-4/99*c^2*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/d/x^9-8/
693*c^4*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/d/x^7-1/110*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^10/(c*x-1)^(1/2)/(c
*x+1)^(1/2)+23/792*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)-113/4158*b*c^5*d^2*(-c^2*d*x
^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/924*b*c^7*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/
2)+2/693*b*c^9*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-8/693*b*c^11*d^2*ln(x)*(-c^2*d*x^2+d)^
(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.58, antiderivative size = 519, normalized size of antiderivative = 1.35, 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} \[ \frac {8 c^{10} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x}+\frac {4 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x^3}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^5}-\frac {5 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {5 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{99 x^9}-\frac {d^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2}}{693 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{924 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {113 b c^5 d^2 \sqrt {d-c^2 d x^2}}{4158 x^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {23 b c^3 d^2 \sqrt {d-c^2 d x^2}}{792 x^8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {c x-1} \sqrt {c x+1}}-\frac {8 b c^{11} d^2 \log (x) \sqrt {d-c^2 d x^2}}{693 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^12,x]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (23*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(7
92*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (113*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(4158*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) + (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(924*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^9*d^2*Sqrt[d - c^2*d*x^
2])/(693*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(231*x^7) +
(c^6*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(231*x^5) + (4*c^8*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c
*x]))/(693*x^3) + (8*c^10*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(693*x) + (5*c^2*d^2*(1 - c*x)*(1 + c*
x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(99*x^9) - (d^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x]))/(11*x^11) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(693*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

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 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 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 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]))

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 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 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]

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^{12}} \, 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^{12}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^5}+\frac {4 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x^3}+\frac {8 c^{10} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x}+\frac {5 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 )}{99 x^9}-\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 )}{11 x^{11}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^3 \left (-63-28 c^2 x^2-8 c^4 x^4\right )}{693 x^{11}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^5}+\frac {4 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x^3}+\frac {8 c^{10} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x}+\frac {5 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 )}{99 x^9}-\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 )}{11 x^{11}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^3 \left (-63-28 c^2 x^2-8 c^4 x^4\right )}{x^{11}} \, dx}{693 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^5}+\frac {4 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x^3}+\frac {8 c^{10} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x}+\frac {5 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 )}{99 x^9}-\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 )}{11 x^{11}}-\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 )^3 \left (-63-28 c^2 x-8 c^4 x^2\right )}{x^6} \, dx,x,x^2\right )}{1386 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^5}+\frac {4 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x^3}+\frac {8 c^{10} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x}+\frac {5 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 )}{99 x^9}-\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 )}{11 x^{11}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {63}{x^6}+\frac {161 c^2}{x^5}-\frac {113 c^4}{x^4}+\frac {3 c^6}{x^3}+\frac {4 c^8}{x^2}+\frac {8 c^{10}}{x}\right ) \, dx,x,x^2\right )}{1386 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^3 d^2 \sqrt {d-c^2 d x^2}}{792 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {113 b c^5 d^2 \sqrt {d-c^2 d x^2}}{4158 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{924 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2}}{693 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^5}+\frac {4 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x^3}+\frac {8 c^{10} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{693 x}+\frac {5 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 )}{99 x^9}-\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 )}{11 x^{11}}-\frac {8 b c^{11} d^2 \sqrt {d-c^2 d x^2} \log (x)}{693 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 165, normalized size = 0.43 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (480 c^2 x^2 (c x-1)^{7/2} \left (2 c^2 x^2+7\right ) (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+7560 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )-b c x \left (960 c^{10} x^{10} \log (x)-240 c^8 x^8-90 c^6 x^6+2260 c^4 x^4-2415 c^2 x^2+756\right )\right )}{83160 x^{11} \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^12,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(7560*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 480*c^2*x^2*(-1 + c*x)^
(7/2)*(1 + c*x)^(7/2)*(7 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) - b*c*x*(756 - 2415*c^2*x^2 + 2260*c^4*x^4 - 90*c^6
*x^6 - 240*c^8*x^8 + 960*c^10*x^10*Log[x])))/(83160*x^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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fricas [A]  time = 0.62, size = 879, normalized size = 2.28 \[ \left [\frac {120 \, {\left (8 \, b c^{12} d^{2} x^{12} - 4 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} + 274 \, b c^{4} d^{2} x^{4} - 224 \, b c^{2} d^{2} x^{2} + 63 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 480 \, {\left (b c^{13} d^{2} x^{13} - b c^{11} d^{2} x^{11}\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 (240 \, b c^{9} d^{2} x^{9} + 90 \, b c^{7} d^{2} x^{7} - {\left (240 \, b c^{9} + 90 \, b c^{7} - 2260 \, b c^{5} + 2415 \, b c^{3} - 756 \, b c\right )} d^{2} x^{11} - 2260 \, b c^{5} d^{2} x^{5} + 2415 \, b c^{3} d^{2} x^{3} - 756 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 120 \, {\left (8 \, a c^{12} d^{2} x^{12} - 4 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} + 274 \, a c^{4} d^{2} x^{4} - 224 \, a c^{2} d^{2} x^{2} + 63 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{83160 \, {\left (c^{2} x^{13} - x^{11}\right )}}, -\frac {960 \, {\left (b c^{13} d^{2} x^{13} - b c^{11} d^{2} x^{11}\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 ) - 120 \, {\left (8 \, b c^{12} d^{2} x^{12} - 4 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} + 274 \, b c^{4} d^{2} x^{4} - 224 \, b c^{2} d^{2} x^{2} + 63 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (240 \, b c^{9} d^{2} x^{9} + 90 \, b c^{7} d^{2} x^{7} - {\left (240 \, b c^{9} + 90 \, b c^{7} - 2260 \, b c^{5} + 2415 \, b c^{3} - 756 \, b c\right )} d^{2} x^{11} - 2260 \, b c^{5} d^{2} x^{5} + 2415 \, b c^{3} d^{2} x^{3} - 756 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 120 \, {\left (8 \, a c^{12} d^{2} x^{12} - 4 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} + 274 \, a c^{4} d^{2} x^{4} - 224 \, a c^{2} d^{2} x^{2} + 63 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{83160 \, {\left (c^{2} x^{13} - x^{11}\right )}}\right ] \]

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^12,x, algorithm="fricas")

[Out]

[1/83160*(120*(8*b*c^12*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 116*b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 -
224*b*c^2*d^2*x^2 + 63*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 480*(b*c^13*d^2*x^13 - b*c^
11*d^2*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)*sq
rt(-d) - d)/(c^2*x^4 - x^2)) + (240*b*c^9*d^2*x^9 + 90*b*c^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 24
15*b*c^3 - 756*b*c)*d^2*x^11 - 2260*b*c^5*d^2*x^5 + 2415*b*c^3*d^2*x^3 - 756*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*s
qrt(c^2*x^2 - 1) + 120*(8*a*c^12*d^2*x^12 - 4*a*c^10*d^2*x^10 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*
d^2*x^4 - 224*a*c^2*d^2*x^2 + 63*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11), -1/83160*(960*(b*c^13*d^2*x^1
3 - b*c^11*d^2*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)) - 120*(8*b*c^12*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 116*b*c^6*d^2*x^6 + 274*b*c^4
*d^2*x^4 - 224*b*c^2*d^2*x^2 + 63*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (240*b*c^9*d^2*x^
9 + 90*b*c^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b*c^3 - 756*b*c)*d^2*x^11 - 2260*b*c^5*d^2*x^
5 + 2415*b*c^3*d^2*x^3 - 756*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 120*(8*a*c^12*d^2*x^12 - 4*a*
c^10*d^2*x^10 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^4 - 224*a*c^2*d^2*x^2 + 63*a*d^2)*sqrt(-c^
2*d*x^2 + d))/(c^2*x^13 - x^11)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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^12,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 1.31, size = 6379, normalized size = 16.57 \[ \text {output too large to display} \]

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^12,x)

[Out]

result too large to display

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maxima [A]  time = 0.80, size = 251, normalized size = 0.65 \[ -\frac {1}{83160} \, {\left (960 \, c^{10} \sqrt {-d} d^{2} \log \relax (x) - \frac {240 \, c^{8} \sqrt {-d} d^{2} x^{8} + 90 \, c^{6} \sqrt {-d} d^{2} x^{6} - 2260 \, c^{4} \sqrt {-d} d^{2} x^{4} + 2415 \, c^{2} \sqrt {-d} d^{2} x^{2} - 756 \, \sqrt {-d} d^{2}}{x^{10}}\right )} b c - \frac {1}{693} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{4}}{d x^{7}} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{9}} + \frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{11}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{693} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{4}}{d x^{7}} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{9}} + \frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{11}}\right )} \]

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^12,x, algorithm="maxima")

[Out]

-1/83160*(960*c^10*sqrt(-d)*d^2*log(x) - (240*c^8*sqrt(-d)*d^2*x^8 + 90*c^6*sqrt(-d)*d^2*x^6 - 2260*c^4*sqrt(-
d)*d^2*x^4 + 2415*c^2*sqrt(-d)*d^2*x^2 - 756*sqrt(-d)*d^2)/x^10)*b*c - 1/693*b*(8*(-c^2*d*x^2 + d)^(7/2)*c^4/(
d*x^7) + 28*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^9) + 63*(-c^2*d*x^2 + d)^(7/2)/(d*x^11))*arccosh(c*x) - 1/693*a*(8
*(-c^2*d*x^2 + d)^(7/2)*c^4/(d*x^7) + 28*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^9) + 63*(-c^2*d*x^2 + d)^(7/2)/(d*x^1
1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^{12}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^12,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^12, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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**12,x)

[Out]

Timed out

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