\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx\) [88]

Optimal result

Integrand size = 27, antiderivative size = 293 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/6*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c^ 
5*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/2*c^4*d^2*x*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))+5/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b 
*arccosh(c*x))/x-1/3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^3-5/4*c^3*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) 
-7/3*b*c^3*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 = 1.26 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {30 b c^3 d^3 x^3 (-1+c x) \text {arccosh}(c x)^2-60 a c^3 d^{5/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+3 b c^3 d^3 x^3 (-1+c x) \cosh (2 \text {arccosh}(c x))-4 d^3 \left (b c x (1-c x)+a \sqrt {\frac {-1+c x}{1+c x}} \left (2-16 c^2 x^2+11 c^4 x^4+3 c^6 x^6\right )-14 b c^3 x^3 (-1+c x) \log (c x)\right )-2 b d^3 (-1+c x) \text {arccosh}(c x) \left (4 \sqrt {\frac {-1+c x}{1+c x}} \left (-1-c x+7 c^2 x^2+7 c^3 x^3\right )+3 c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )}{24 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^4,x]
 

Output:

(30*b*c^3*d^3*x^3*(-1 + c*x)*ArcCosh[c*x]^2 - 60*a*c^3*d^(5/2)*x^3*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))] + 3*b*c^3*d^3*x^3*(-1 + c*x)*Cosh[2*ArcCosh[c*x]] 
 - 4*d^3*(b*c*x*(1 - c*x) + a*Sqrt[(-1 + c*x)/(1 + c*x)]*(2 - 16*c^2*x^2 + 
 11*c^4*x^4 + 3*c^6*x^6) - 14*b*c^3*x^3*(-1 + c*x)*Log[c*x]) - 2*b*d^3*(-1 
 + c*x)*ArcCosh[c*x]*(4*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 - c*x + 7*c^2*x^2 + 
 7*c^3*x^3) + 3*c^3*x^3*Sinh[2*ArcCosh[c*x]]))/(24*x^3*Sqrt[(-1 + c*x)/(1 
+ c*x)]*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.06, 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, 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.

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

\(\Big \downarrow \) 6343

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6343

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6308

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

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^4,x]
 

Output:

-1/3*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^3 - (5*c^2*d*(-(((d - 
c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x) - 3*c^2*d*(-1/4*(b*c*x^2*Sqrt[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])) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/2*(c^2*x^2 
) + Log[x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/3 + (b*c*d^2*Sqrt[d - c^2*d* 
x^2]*(-x^(-2) + c^4*x^2 - 2*c^2*Log[x^2]))/(6*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 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 = 0.41 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-56 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\)	\(340\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-56 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\)	\(340\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*a/d/x^3*(-c^2*d*x^2+d)^(7/2)+4/3*a*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+4/3*a 
*c^4*x*(-c^2*d*x^2+d)^(5/2)+5/3*a*c^4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d 
^2*x*(-c^2*d*x^2+d)^(1/2)+5/2*a*c^4*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) 
*x/(-c^2*d*x^2+d)^(1/2))-1/24*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+ 
1)^(1/2)/x^3*(-12*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*x^4*c^4+6*c^5*x 
^5+30*arccosh(c*x)^2*c^3*x^3-56*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c 
^2*x^2-56*c^3*x^3*arccosh(c*x)+56*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2 
)*x^3*c^3-3*c^3*x^3+8*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*d^2
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^4,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^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**4,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^4,x, algorithm="maxima 
")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^4,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^4,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a \,c^{3} x^{3}+3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+14 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a +6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{4}}d x \right ) b \,x^{3}-12 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{2}}d x \right ) b \,c^{2} x^{3}+6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )d x \right ) b \,c^{4} x^{3}\right )}{6 x^{3}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))/x^4,x)
 

Output:

(sqrt(d)*d**2*(15*asin(c*x)*a*c**3*x**3 + 3*sqrt( - c**2*x**2 + 1)*a*c**4* 
x**4 + 14*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a 
+ 6*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**4,x)*b*x**3 - 12*int((sqrt( 
 - c**2*x**2 + 1)*acosh(c*x))/x**2,x)*b*c**2*x**3 + 6*int(sqrt( - c**2*x** 
2 + 1)*acosh(c*x),x)*b*c**4*x**3))/(6*x**3)