\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx\) [77]

Optimal result

Integrand size = 27, antiderivative size = 247 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2}}{35 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{70 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}+\frac {2 b c^7 d \sqrt {d-c^2 d x^2} \log (x)}{35 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^7-2/35*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^5-1/42
*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/35*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2
)/(c*x+1)^(1/2)-1/70*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/35*b*c^7*d*ln(x)*(-c^2*d*x
^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {277, 270, 5922, 12, 457, 77} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^7 d \log (x) \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{70 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2}}{35 x^4 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^8,x]

[Out]

-1/42*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*Sqrt[d - c^2*d*x^2])/(35*x^4
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(70*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c
^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(35*d*x^5
) + (2*b*c^7*d*Sqrt[d - c^2*d*x^2]*Log[x])/(35*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 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 270

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

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {d \left (-5-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 x^7} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-5-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{x^7} \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (-5-2 c^2 x\right ) \left (1-c^2 x\right )^2}{x^4} \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {5}{x^4}+\frac {8 c^2}{x^3}-\frac {c^4}{x^2}-\frac {2 c^6}{x}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2}}{35 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{70 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}+\frac {2 b c^7 d \sqrt {d-c^2 d x^2} \log (x)}{35 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (30 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+12 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+b c x \left (5-12 c^2 x^2+3 c^4 x^4-12 c^6 x^6 \log (x)\right )\right )}{210 x^7 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^8,x]

[Out]

-1/210*(d*Sqrt[d - c^2*d*x^2]*(30*(-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)*(a + b*ArcCosh[c*x]) + b*c*x*(5 - 12*c^2*x^2 + 3*c^4*x^4 - 12*c^6*x^6*Log[x])))/(x^7*S
qrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3144\) vs. \(2(207)=414\).

Time = 1.17 (sec) , antiderivative size = 3145, normalized size of antiderivative = 12.73

method	result	size
default	\(\text {Expression too large to display}\)	\(3145\)
parts	\(\text {Expression too large to display}\)	\(3145\)

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x,method=_RETURNVERBOSE)

[Out]

142/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^7/(c*x+1)
/(c*x-1)*c^14-2/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*
x^11*c^18+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^9*c
^16+26/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^7*c^14
-116/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^5*c^12+2
0/21*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^3*c^10-5/21*
b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x*c^8-164/5*b*(-d*(
c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^5/(c*x+1)/(c*x-1)*arccos
h(c*x)*c^12+52/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^
3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+1966/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154
*c^4*x^4-105*c^2*x^2+25)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-3272/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-3
5*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+472/7*b*(-d*(c^2*x^2-1))^(
1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-2*
b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^8/(c*x+1)^(1/2)/(
c*x-1)^(1/2)*arccosh(c*x)*c^15-4*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-10
5*c^2*x^2+25)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13+44/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-
35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11-6*b*(-d*(c
^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^2/(c*x+1)^(1/2)/(c*x-1)^(
1/2)*arccosh(c*x)*c^9+2*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2
+25)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^17+a*(-1/7/d/x^7*(-c^2*d*x^2+d)^(5/2)-2/35*c^2/d/x^5*(-c^
2*d*x^2+d)^(5/2))+359/30*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^
2+25)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7-4/35*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d
*c^7+2/35*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*d*c^7
-1/21*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^9/(c*x+1)/(
c*x-1)*c^16+2/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^
13/(c*x+1)/(c*x-1)*c^20-9/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^
2*x^2+25)*x^11/(c*x+1)/(c*x-1)*c^18+5/2*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4
*x^4-105*c^2*x^2+25)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13-11/6*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8
*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11+10/7*b*(-d*(c^2*x^2-1))^(1/2)
*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^
7-161/30*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^2/(c*x+1
)^(1/2)/(c*x-1)^(1/2)*c^9-421/42*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-10
5*c^2*x^2+25)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5+55/14*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70
*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-25/42*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*
c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c-1/2*b*(-d*(c^2*x
^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)
*c^15-72/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^5/(c*
x+1)/(c*x-1)*c^12+25/21*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2
+25)*x^3/(c*x+1)/(c*x-1)*c^10-5/21*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-
105*c^2*x^2+25)*x/(c*x+1)/(c*x-1)*c^8+25/7*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*
c^4*x^4-105*c^2*x^2+25)/x^7/(c*x+1)/(c*x-1)*arccosh(c*x)-2*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8
-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^11/(c*x+1)/(c*x-1)*arccosh(c*x)*c^18+3*b*(-d*(c^2*x^2-1))^(1/2)*d/(3
5*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c^16+12*b*(-d*(
c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^7/(c*x+1)/(c*x-1)*arccos
h(c*x)*c^14-170/7*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)/x
^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.62 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\left [-\frac {6 \, {\left (2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 5 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (b c^{9} d x^{9} - b c^{7} d x^{7}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (3 \, b c^{5} d x^{5} - {\left (3 \, b c^{5} - 12 \, b c^{3} + 5 \, b c\right )} d x^{7} - 12 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 6 \, {\left (2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 9 \, a c^{4} d x^{4} + 13 \, a c^{2} d x^{2} - 5 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{210 \, {\left (c^{2} x^{9} - x^{7}\right )}}, \frac {12 \, {\left (b c^{9} d x^{9} - b c^{7} d x^{7}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 6 \, {\left (2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 5 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (3 \, b c^{5} d x^{5} - {\left (3 \, b c^{5} - 12 \, b c^{3} + 5 \, b c\right )} d x^{7} - 12 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left (2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 9 \, a c^{4} d x^{4} + 13 \, a c^{2} d x^{2} - 5 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{210 \, {\left (c^{2} x^{9} - x^{7}\right )}}\right ] \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="fricas")

[Out]

[-1/210*(6*(2*b*c^8*d*x^8 - b*c^6*d*x^6 - 9*b*c^4*d*x^4 + 13*b*c^2*d*x^2 - 5*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x
+ sqrt(c^2*x^2 - 1)) - 6*(b*c^9*d*x^9 - b*c^7*d*x^7)*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)) + (3*b*c^5*d*x^5 - (3*b*c^5 - 12*b*c^3 +
5*b*c)*d*x^7 - 12*b*c^3*d*x^3 + 5*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 6*(2*a*c^8*d*x^8 - a*c^6*
d*x^6 - 9*a*c^4*d*x^4 + 13*a*c^2*d*x^2 - 5*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7), 1/210*(12*(b*c^9*d*x^9
- b*c^7*d*x^7)*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*(2*b*c^8*d*x^8 - b*c^6*d*x^6 - 9*b*c^4*d*x^4 + 13*b*c^2*d*x^2 - 5*b*d)*sqrt(-c^2*d*x^2 + d)*lo
g(c*x + sqrt(c^2*x^2 - 1)) - (3*b*c^5*d*x^5 - (3*b*c^5 - 12*b*c^3 + 5*b*c)*d*x^7 - 12*b*c^3*d*x^3 + 5*b*c*d*x)
*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*(2*a*c^8*d*x^8 - a*c^6*d*x^6 - 9*a*c^4*d*x^4 + 13*a*c^2*d*x^2 - 5*
a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Timed out} \]

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**8,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {1}{210} \, {\left (12 \, c^{6} \sqrt {-d} d \log \left (x\right ) - \frac {3 \, c^{4} \sqrt {-d} d x^{4} - 12 \, c^{2} \sqrt {-d} d x^{2} + 5 \, \sqrt {-d} d}{x^{6}}\right )} b c - \frac {1}{35} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{5}} + \frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{35} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{5}} + \frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="maxima")

[Out]

1/210*(12*c^6*sqrt(-d)*d*log(x) - (3*c^4*sqrt(-d)*d*x^4 - 12*c^2*sqrt(-d)*d*x^2 + 5*sqrt(-d)*d)/x^6)*b*c - 1/3
5*b*(2*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^5) + 5*(-c^2*d*x^2 + d)^(5/2)/(d*x^7))*arccosh(c*x) - 1/35*a*(2*(-c^2*d
*x^2 + d)^(5/2)*c^2/(d*x^5) + 5*(-c^2*d*x^2 + d)^(5/2)/(d*x^7))

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^8} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^8,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^8, x)