3.1.92 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx\) [92]

3.1.92.1 Optimal result

Integrand size = 27, antiderivative size = 293 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {11 b c^3 d^2 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}+\frac {c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {c^5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^5 d^2 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {-1+c x} \sqrt {1+c x}} \]

output

1/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^3-1/5*(-c^2*d*x^2+d)^( 
5/2)*(a+b*arccosh(c*x))/x^5-c^4*d^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2 
)/x-1/20*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+11/3 
0*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*c^5*d 
^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/(c*x-1)^(1/2)/(c*x+1)^(1/2) 
+23/15*b*c^5*d^2*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

3.1.92.2 Mathematica [A] (warning: unable to verify)

Time = 2.94 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\frac {d^2 \left (8 a d \sqrt {\frac {-1+c x}{1+c x}} \left (-1+c^2 x^2\right ) \left (3-11 c^2 x^2+23 c^4 x^4\right )+120 a c^5 \sqrt {d} x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+40 b c^2 d x^2 (1-c x) \left (c x-2 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)+2 c^3 x^3 \log (c x)\right )-60 b c^4 d x^4 (1-c x) \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x)^2+2 \log (c x)\right )\right )-b d (1-c x) \left (20 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)+\cosh (5 \text {arccosh}(c x)) \log (c x)+\cosh (3 \text {arccosh}(c x)) (-1+5 \log (c x))+c x (3+10 \log (c x))-5 \text {arccosh}(c x) \sinh (3 \text {arccosh}(c x))-\text {arccosh}(c x) \sinh (5 \text {arccosh}(c x))\right )\right )}{120 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]

input

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]
 

output

(d^2*(8*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + c^2*x^2)*(3 - 11*c^2*x^2 + 23 
*c^4*x^4) + 120*a*c^5*Sqrt[d]*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2* 
d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 40*b*c 
^2*d*x^2*(1 - c*x)*(c*x - 2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCo 
sh[c*x] + 2*c^3*x^3*Log[c*x]) - 60*b*c^4*d*x^4*(1 - c*x)*(2*Sqrt[(-1 + c*x 
)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - 
 b*d*(1 - c*x)*(20*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + Cos 
h[5*ArcCosh[c*x]]*Log[c*x] + Cosh[3*ArcCosh[c*x]]*(-1 + 5*Log[c*x]) + c*x* 
(3 + 10*Log[c*x]) - 5*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - ArcCosh[c*x]*Sin 
h[5*ArcCosh[c*x]])))/(120*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^ 
2])
 

3.1.92.3 Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6343, 82, 243, 49, 2009, 6343, 25, 82, 244, 2009, 6339, 14, 6308}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx\)

\(\Big \downarrow \) 6343

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2}{x^5}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}\)

\(\Big \downarrow \) 82

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^5}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}\)

\(\Big \downarrow \) 243

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^6}dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}\)

\(\Big \downarrow \) 49

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {c^4}{x^2}-\frac {2 c^2}{x^4}+\frac {1}{x^6}\right )dx^2}{10 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^4}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6343

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1)}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1)}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 82

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 244

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right )dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6339

\(\displaystyle -c^2 d \left (c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 14

\(\displaystyle -c^2 d \left (c^2 (-d) \left (\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 x^5}-c^2 d \left (c^2 (-d) \left (\frac {c \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {c x-1} \sqrt {c x+1}}\)

input

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]
 

output

-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^5 - c^2*d*(-1/3*((d - 
c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^3 + (b*c*d*Sqrt[d - c^2*d*x^2]*(- 
1/2*1/x^2 - c^2*Log[x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - c^2*d*(-((Sqrt 
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x) + (c*Sqrt[d - c^2*d*x^2]*(a + b*A 
rcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d - c^2*d*x 
^2]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]))) + (b*c*d^2*Sqrt[d - c^2*d*x^2 
]*(-1/2*1/x^4 + (2*c^2)/x^2 + c^4*Log[x^2]))/(10*Sqrt[-1 + c*x]*Sqrt[1 + c 
*x])
 

3.1.92.3.1 Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6339

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

rule 6343

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

3.1.92.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2428\) vs. \(2(251)=502\).

Time = 1.22 (sec) , antiderivative size = 2429, normalized size of antiderivative = 8.29

method	result	size
default	\(\text {Expression too large to display}\)	\(2429\)
parts	\(\text {Expression too large to display}\)	\(2429\)

input

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x,method=_RETURNVERBOSE)
 

output

1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*d^ 
2*c^5+23/15*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^2*c^5-46/15*b*(-d*(c^2*x^2-1))^(1/2)/(c 
*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d^2*c^5-175/4*b*(-d*(c^2*x^2-1))^(1 
/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/(c*x+1)^(1/2)/ 
(c*x-1)^(1/2)*c^5+2/15*a*c^2/d/x^3*(-c^2*d*x^2+d)^(7/2)-1329/4*b*(-d*(c^2* 
x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^4/ 
(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9+1889/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035 
*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^2/(c*x+1)^(1/2)/(c*x-1)^( 
1/2)*c^7+141/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325 
*c^4*x^4-75*c^2*x^2+9)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-5819/30*b*(-d*( 
c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)* 
x^9/(c*x+1)/(c*x-1)*c^14+18791/60*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x 
^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^7/(c*x+1)/(c*x-1)*c^12-943/6*b* 
(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^ 
2+9)*x^5/(c*x+1)/(c*x-1)*c^10+207/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8 
*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^3/(c*x+1)/(c*x-1)*c^8-69/20*b 
*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x 
^2+9)*x/(c*x+1)/(c*x-1)*c^6+9/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8 
-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)...
 

3.1.92.5 Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]

input

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="fricas 
")
 

output

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c 
^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/x^6, x)
 

3.1.92.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\text {Timed out} \]

input

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**6,x)
 

output

Timed out
 

3.1.92.7 Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]

input

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="maxima 
")
 

output

-1/15*(10*(-c^2*d*x^2 + d)^(3/2)*c^6*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^6*d^2 
*x + 15*c^5*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^4/x - 2*(-c^2 
*d*x^2 + d)^(7/2)*c^2/(d*x^3) + 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^5))*a + b*in 
tegrate((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^6, 
 x)
 

3.1.92.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\text {Exception raised: TypeError} \]

input

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="giac")
 

output

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

3.1.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^6} \,d x \]

input

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^6,x)
 

output

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^6, x)