\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^2} \, dx\) [87]

Optimal result

Integrand size = 27, antiderivative size = 326 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {b c^3 d^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 x^4 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{16} b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2} \sqrt {d-c^2 d x^2}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x}+\frac {15 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/16*b*c^3*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/4*b 
*c^5*d^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/16*b*c*d^2 
*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)-15/8*c^2*d^2*x*(-c^2*d*x 
^2+d)^(1/2)*(a+b*arccosh(c*x))-5/4*c^2*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcco 
sh(c*x))-(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x+15/16*c*d^2*(-c^2*d*x^2 
+d)^(1/2)*(a+b*arccosh(c*x))^2/b/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c*d^2*(-c^2 
*d*x^2+d)^(1/2)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.47 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{128} d^2 \left (\frac {16 a \sqrt {d-c^2 d x^2} \left (-8-9 c^2 x^2+2 c^4 x^4\right )}{x}+240 a c \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+64 b c \sqrt {d-c^2 d x^2} \left (-\frac {2 \text {arccosh}(c x)}{c x}+\frac {\text {arccosh}(c x)^2+2 \log (c x)}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {32 b c \sqrt {d-c^2 d x^2} \left (2 \text {arccosh}(c x)^2+\cosh (2 \text {arccosh}(c x))-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {b c \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \] Input:

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

Output:

(d^2*((16*a*Sqrt[d - c^2*d*x^2]*(-8 - 9*c^2*x^2 + 2*c^4*x^4))/x + 240*a*c* 
Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 64*b* 
c*Sqrt[d - c^2*d*x^2]*((-2*ArcCosh[c*x])/(c*x) + (ArcCosh[c*x]^2 + 2*Log[c 
*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (32*b*c*Sqrt[d - c^2*d*x^2] 
*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[ 
c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*c*Sqrt[d - c^2*d*x^2]* 
(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c 
*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/128
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.97, 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, 6312, 25, 82, 244, 2009, 6310, 15, 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.

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

\(\Big \downarrow \) 6343

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6312

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6308

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

Input:

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

Output:

-(((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x) - 5*c^2*d*(-1/4*(b*c*d*S 
qrt[d - c^2*d*x^2]*(x^2/2 - (c^2*x^4)/4))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 
 (x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/4 + (3*d*(-1/4*(b*c*x^2*Sq 
rt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*Sqrt[d - c^2*d*x^2] 
*(a + b*ArcCosh[c*x]))/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4 
*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/4) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-2 
*c^2*x^2 + (c^4*x^4)/2 + Log[x^2]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 6310

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[x*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n 
, 0]
 

rule 6312

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p 
)]   Int[x*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 
0] && GtQ[p, 0]
 

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]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a \,c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a \,c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a \,c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}-8 c^{5} x^{5}-144 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+72 c^{3} x^{3}+120 \operatorname {arccosh}\left (c x \right )^{2} c x -128 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-128 c x \,\operatorname {arccosh}\left (c x \right )+128 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \right ) d^{2}}{128 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\)	\(303\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a \,c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a \,c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a \,c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}-8 c^{5} x^{5}-144 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}+72 c^{3} x^{3}+120 \operatorname {arccosh}\left (c x \right )^{2} c x -128 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-128 c x \,\operatorname {arccosh}\left (c x \right )+128 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \right ) d^{2}}{128 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\)	\(303\)

Input:

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

Output:

-a/d/x*(-c^2*d*x^2+d)^(7/2)-a*c^2*x*(-c^2*d*x^2+d)^(5/2)-5/4*a*c^2*d*x*(-c 
^2*d*x^2+d)^(3/2)-15/8*a*c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)-15/8*a*c^2*d^3/(c^ 
2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/128*b*(-d*(c^2*x 
^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/x*(32*(c*x-1)^(1/2)*(c*x+1)^(1/2) 
*arccosh(c*x)*x^4*c^4-8*c^5*x^5-144*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/ 
2)*c^2*x^2+72*c^3*x^3+120*arccosh(c*x)^2*c*x-128*arccosh(c*x)*(c*x-1)^(1/2 
)*(c*x+1)^(1/2)-128*c*x*arccosh(c*x)+128*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^( 
1/2))^2)*x*c-33*c*x)*d^2
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^2,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^2, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*(10*(-c^2*d*x^2 + d)^(3/2)*c^2*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^2*d^2* 
x + 15*c*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)/x)*a + b*integrate 
((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^2, x)
 

Giac [F(-2)]

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

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^2,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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {\sqrt {d}\, d^{2} \left (-15 \mathit {asin} \left (c x \right ) a c x +2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-9 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a +8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{2}}d x \right ) b x +8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{4} x -16 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{2} x \right )}{8 x} \] Input:

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

Output:

(sqrt(d)*d**2*( - 15*asin(c*x)*a*c*x + 2*sqrt( - c**2*x**2 + 1)*a*c**4*x** 
4 - 9*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 8*sqrt( - c**2*x**2 + 1)*a + 8* 
int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**2,x)*b*x + 8*int(sqrt( - c**2*x 
**2 + 1)*acosh(c*x)*x**2,x)*b*c**4*x - 16*int(sqrt( - c**2*x**2 + 1)*acosh 
(c*x),x)*b*c**2*x))/(8*x)