\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx\) [64]

Optimal result

Integrand size = 27, antiderivative size = 279 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{140 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^5 \sqrt {d-c^2 d x^2}}{105 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{105 d x^3}-\frac {8 b c^7 \sqrt {d-c^2 d x^2} \log (x)}{105 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/d/x^7-4/35*c^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/d/x^5-8/10
5*c^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/d/x^3-1/42*b*c*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1
/2)+1/140*b*c^3*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/105*b*c^5*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x
-1)^(1/2)/(c*x+1)^(1/2)-8/105*b*c^7*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {277, 270, 5922, 12, 14} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{105 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {8 b c^7 \log (x) \sqrt {d-c^2 d x^2}}{105 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^5 \sqrt {d-c^2 d x^2}}{105 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{140 x^4 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

-1/42*(b*c*Sqrt[d - c^2*d*x^2])/(x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d
*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (4*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(35*d*x^5) -
(8*c^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(105*d*x^3) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqr
t[-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 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 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 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 )^{3/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{105 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{105 x^7} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{105 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{x^7} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{105 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {15}{x^7}+\frac {3 c^2}{x^5}+\frac {4 c^4}{x^3}+\frac {8 c^6}{x}\right ) \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{140 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^5 \sqrt {d-c^2 d x^2}}{105 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{105 d x^3}-\frac {8 b c^7 \sqrt {d-c^2 d x^2} \log (x)}{105 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (60 (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))+16 c^2 x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (3+2 c^2 x^2\right ) (a+b \text {arccosh}(c x))-b c x \left (10-3 c^2 x^2-8 c^4 x^4+32 c^6 x^6 \log (x)\right )\right )}{420 x^7 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(60*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + 16*c^2*x^2*(-1 + c*x)^(3/2)*(
1 + c*x)^(3/2)*(3 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) - b*c*x*(10 - 3*c^2*x^2 - 8*c^4*x^4 + 32*c^6*x^6*Log[x])))
/(420*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2536\) vs. \(2(235)=470\).

Time = 1.17 (sec) , antiderivative size = 2537, normalized size of antiderivative = 9.09

method	result	size
default	\(\text {Expression too large to display}\)	\(2537\)
parts	\(\text {Expression too large to display}\)	\(2537\)

[In]

int((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

a*(-1/7/d/x^7*(-c^2*d*x^2+d)^(3/2)+4/7*c^2*(-1/5/d/x^5*(-c^2*d*x^2+d)^(3/2)-2/15*c^2/d/x^3*(-c^2*d*x^2+d)^(3/2
)))+8*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^6/(c*x-1)^(1/2)/(c*x+1)^
(1/2)*arccosh(c*x)*c^13+8/5*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^4/
(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^11+24*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-
315*c^2*x^2+225)*x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^9-64/3*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-1
05*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^15-56/3*b*(-d*(c^2*x^2-1
))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7/(c*x-1)/(c*x+1)*arccosh(c*x)*c^14-4/15*b*(-d
*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5/(c*x-1)/(c*x+1)*arccosh(c*x)*c^12
-351/5*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3/(c*x-1)/(c*x+1)*arcco
sh(c*x)*c^10+3057/35*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x/(c*x-1)/(
c*x+1)*arccosh(c*x)*c^8-594/35*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x
/(c*x-1)/(c*x+1)*arccosh(c*x)*c^6+128/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2
*x^2+225)*x^11*c^18+16/15*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9*c^
16-88/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7*c^14-302/105*b*(-d
*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5*c^12-10/7*b*(-d*(c^2*x^2-1))^(1/2
)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3*c^10+20/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105
*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x*c^8-128/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^
4-315*c^2*x^2+225)*x^13/(c*x-1)/(c*x+1)*c^20+16/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x
^4-315*c^2*x^2+225)*x^11/(c*x-1)/(c*x+1)*c^18-120/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x
^4-315*c^2*x^2+225)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^7+342/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-1
05*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^3/(c*x-1)/(c*x+1)*arccosh(c*x)*c^4-585/7*b*(-d*(c^2*x^2-1))^(1/2)/(28
0*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^5/(c*x-1)/(c*x+1)*arccosh(c*x)*c^2+64/3*b*(-d*(c^2*x^2-1))
^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9/(c*x-1)/(c*x+1)*arccosh(c*x)*c^16+16/105*b*(-d
*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^7-8/105*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)*c^7-73/20*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^
6*x^6-21*c^4*x^4-315*c^2*x^2+225)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^7+40/21*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-
105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9/(c*x-1)/(c*x+1)*c^16+214/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8
-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7/(c*x-1)/(c*x+1)*c^14-152/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^
8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5/(c*x-1)/(c*x+1)*c^12-30/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-
105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3/(c*x-1)/(c*x+1)*c^10+20/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-10
5*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x/(c*x-1)/(c*x+1)*c^8+225/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^
6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^7/(c*x-1)/(c*x+1)*arccosh(c*x)+16/3*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-
105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^13-75/14*b*(-d*(c^2*x^2-1))^(1/2)/(2
80*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c-469/60*b*(-d*(c^2*x^2-1))
^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^9+71/28*b*(-d*(c
^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^5+255/
28*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/
2)*c^3

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.20 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\left [\frac {4 \, {\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 16 \, {\left (b c^{9} x^{9} - b c^{7} x^{7}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (8 \, b c^{5} x^{5} - {\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 4 \, {\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} + 15 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{420 \, {\left (c^{2} x^{9} - x^{7}\right )}}, -\frac {32 \, {\left (b c^{9} x^{9} - b c^{7} x^{7}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 4 \, {\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} x^{5} - {\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} + 15 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{420 \, {\left (c^{2} x^{9} - x^{7}\right )}}\right ] \]

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x, algorithm="fricas")

[Out]

[1/420*(4*(8*b*c^8*x^8 - 4*b*c^6*x^6 - b*c^4*x^4 - 18*b*c^2*x^2 + 15*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^
2*x^2 - 1)) + 16*(b*c^9*x^9 - b*c^7*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)*sq
rt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) + (8*b*c^5*x^5 - (8*b*c^5 + 3*b*c^3 - 10*b*c)*x^7 + 3
*b*c^3*x^3 - 10*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 4*(8*a*c^8*x^8 - 4*a*c^6*x^6 - a*c^4*x^4 - 18*
a*c^2*x^2 + 15*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7), -1/420*(32*(b*c^9*x^9 - b*c^7*x^7)*sqrt(d)*arctan(sqr
t(-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)) - 4*(8*b*c^8*x^8 - 4*
b*c^6*x^6 - b*c^4*x^4 - 18*b*c^2*x^2 + 15*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5*x^5
- (8*b*c^5 + 3*b*c^3 - 10*b*c)*x^7 + 3*b*c^3*x^3 - 10*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 4*(8*a*c
^8*x^8 - 4*a*c^6*x^6 - a*c^4*x^4 - 18*a*c^2*x^2 + 15*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]

Sympy [F(-1)]

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

[In]

integrate((a+b*acosh(c*x))*(-c**2*d*x**2+d)**(1/2)/x**8,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {1}{420} \, {\left (32 \, c^{6} \sqrt {-d} \log \left (x\right ) - \frac {8 \, c^{4} \sqrt {-d} x^{4} + 3 \, c^{2} \sqrt {-d} x^{2} - 10 \, \sqrt {-d}}{x^{6}}\right )} b c - \frac {1}{105} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4}}{d x^{3}} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{5}} + \frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{7}}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{105} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4}}{d x^{3}} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{5}} + \frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{7}}\right )} a \]

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x, algorithm="maxima")

[Out]

-1/420*(32*c^6*sqrt(-d)*log(x) - (8*c^4*sqrt(-d)*x^4 + 3*c^2*sqrt(-d)*x^2 - 10*sqrt(-d))/x^6)*b*c - 1/105*(8*(
-c^2*d*x^2 + d)^(3/2)*c^4/(d*x^3) + 12*(-c^2*d*x^2 + d)^(3/2)*c^2/(d*x^5) + 15*(-c^2*d*x^2 + d)^(3/2)/(d*x^7))
*b*arccosh(c*x) - 1/105*(8*(-c^2*d*x^2 + d)^(3/2)*c^4/(d*x^3) + 12*(-c^2*d*x^2 + d)^(3/2)*c^2/(d*x^5) + 15*(-c
^2*d*x^2 + d)^(3/2)/(d*x^7))*a

Giac [F(-2)]

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

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^8} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^8,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^8, x)