\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx\) [92]

Optimal result

Integrand size = 27, antiderivative size = 385 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^3 d^2 \sqrt {d-c^2 d x^2}}{792 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {113 b c^5 d^2 \sqrt {d-c^2 d x^2}}{4158 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{924 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2}}{693 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}-\frac {8 b c^{11} d^2 \sqrt {d-c^2 d x^2} \log (x)}{693 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/110*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^10/(c*x-1)^(1/2)/(c*x+1)^(1/2)+23/79 
2*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)-113/4158* 
b*c^5*d^2*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/924*b*c^7 
*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/693*b*c^9*d^2* 
(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/11*(-c^2*d*x^2+d)^( 
7/2)*(a+b*arccosh(c*x))/d/x^11-4/99*c^2*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh( 
c*x))/d/x^9-8/693*c^4*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/d/x^7-8/693* 
b*c^11*d^2*(-c^2*d*x^2+d)^(1/2)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.43 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (7560 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+480 c^2 x^2 (-1+c x)^{7/2} (1+c x)^{7/2} \left (7+2 c^2 x^2\right ) (a+b \text {arccosh}(c x))-b c x \left (756-2415 c^2 x^2+2260 c^4 x^4-90 c^6 x^6-240 c^8 x^8+960 c^{10} x^{10} \log (x)\right )\right )}{83160 x^{11} \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^12,x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(7560*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*Arc 
Cosh[c*x]) + 480*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(7 + 2*c^2*x^2)* 
(a + b*ArcCosh[c*x]) - b*c*x*(756 - 2415*c^2*x^2 + 2260*c^4*x^4 - 90*c^6*x 
^6 - 240*c^8*x^8 + 960*c^10*x^10*Log[x])))/(83160*x^11*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 1578, 1195, 2009}

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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx\)

\(\Big \downarrow \) 6337

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (8 c^4 x^4+28 c^2 x^2+63\right )}{693 x^{11}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3 \left (8 c^4 x^4+28 c^2 x^2+63\right )}{x^{11}}dx}{693 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3 \left (8 c^4 x^4+28 c^2 x^2+63\right )}{x^{12}}dx^2}{1386 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (-\frac {8 c^{10}}{x^2}-\frac {4 c^8}{x^4}-\frac {3 c^6}{x^6}+\frac {113 c^4}{x^8}-\frac {161 c^2}{x^{10}}+\frac {63}{x^{12}}\right )dx^2}{1386 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-8 c^{10} \log \left (x^2\right )+\frac {4 c^8}{x^2}+\frac {3 c^6}{2 x^4}-\frac {113 c^4}{3 x^6}+\frac {161 c^2}{4 x^8}-\frac {63}{5 x^{10}}\right )}{1386 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^12,x]
 

Output:

-1/11*((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(d*x^11) - (4*c^2*(d - 
c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(99*d*x^9) - (8*c^4*(d - c^2*d*x^2) 
^(7/2)*(a + b*ArcCosh[c*x]))/(693*d*x^7) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(- 
63/(5*x^10) + (161*c^2)/(4*x^8) - (113*c^4)/(3*x^6) + (3*c^6)/(2*x^4) + (4 
*c^8)/x^2 - 8*c^10*Log[x^2]))/(1386*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1195

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

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

rule 6337

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]}, Simp[(a + b*ArcCo 
sh[c*x])   u, x] - Simp[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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6381\) vs. \(2(325)=650\).

Time = 0.63 (sec) , antiderivative size = 6382, normalized size of antiderivative = 16.58

method	result	size
default	\(\text {Expression too large to display}\)	\(6382\)
parts	\(\text {Expression too large to display}\)	\(6382\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^12,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.29 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx =\text {Too large to display} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^12,x, algorithm="frica 
s")
 

Output:

[1/83160*(120*(8*b*c^12*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 116 
*b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 - 224*b*c^2*d^2*x^2 + 63*b*d^2)*sqrt(-c 
^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 480*(b*c^13*d^2*x^13 - b*c^11 
*d^2*x^11)*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)) + (240*b*c^ 
9*d^2*x^9 + 90*b*c^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b 
*c^3 - 756*b*c)*d^2*x^11 - 2260*b*c^5*d^2*x^5 + 2415*b*c^3*d^2*x^3 - 756*b 
*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 120*(8*a*c^12*d^2*x^12 
- 4*a*c^10*d^2*x^10 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^ 
4 - 224*a*c^2*d^2*x^2 + 63*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11), 
 -1/83160*(960*(b*c^13*d^2*x^13 - b*c^11*d^2*x^11)*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)) - 120*(8*b*c^12*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 1 
16*b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 - 224*b*c^2*d^2*x^2 + 63*b*d^2)*sqrt( 
-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (240*b*c^9*d^2*x^9 + 90*b*c 
^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b*c^3 - 756*b*c)*d^ 
2*x^11 - 2260*b*c^5*d^2*x^5 + 2415*b*c^3*d^2*x^3 - 756*b*c*d^2*x)*sqrt(-c^ 
2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 120*(8*a*c^12*d^2*x^12 - 4*a*c^10*d^2*x^1 
0 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^4 - 224*a*c^2*d^2* 
x^2 + 63*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**12,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.65 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {1}{83160} \, {\left (960 \, c^{10} \sqrt {-d} d^{2} \log \left (x\right ) - \frac {240 \, c^{8} \sqrt {-d} d^{2} x^{8} + 90 \, c^{6} \sqrt {-d} d^{2} x^{6} - 2260 \, c^{4} \sqrt {-d} d^{2} x^{4} + 2415 \, c^{2} \sqrt {-d} d^{2} x^{2} - 756 \, \sqrt {-d} d^{2}}{x^{10}}\right )} b c - \frac {1}{693} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{4}}{d x^{7}} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{9}} + \frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{11}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{693} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{4}}{d x^{7}} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{9}} + \frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{11}}\right )} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^12,x, algorithm="maxim 
a")
 

Output:

-1/83160*(960*c^10*sqrt(-d)*d^2*log(x) - (240*c^8*sqrt(-d)*d^2*x^8 + 90*c^ 
6*sqrt(-d)*d^2*x^6 - 2260*c^4*sqrt(-d)*d^2*x^4 + 2415*c^2*sqrt(-d)*d^2*x^2 
 - 756*sqrt(-d)*d^2)/x^10)*b*c - 1/693*b*(8*(-c^2*d*x^2 + d)^(7/2)*c^4/(d* 
x^7) + 28*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^9) + 63*(-c^2*d*x^2 + d)^(7/2)/( 
d*x^11))*arccosh(c*x) - 1/693*a*(8*(-c^2*d*x^2 + d)^(7/2)*c^4/(d*x^7) + 28 
*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^9) + 63*(-c^2*d*x^2 + d)^(7/2)/(d*x^11))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^12,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^{12}} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^12,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^12, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\frac {\sqrt {d}\, d^{2} \left (8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{10} x^{10}+4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}+3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-113 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+161 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-63 \sqrt {-c^{2} x^{2}+1}\, a +693 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{12}}d x \right ) b \,x^{11}-1386 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{10}}d x \right ) b \,c^{2} x^{11}+693 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{8}}d x \right ) b \,c^{4} x^{11}\right )}{693 x^{11}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))/x^12,x)
 

Output:

(sqrt(d)*d**2*(8*sqrt( - c**2*x**2 + 1)*a*c**10*x**10 + 4*sqrt( - c**2*x** 
2 + 1)*a*c**8*x**8 + 3*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 113*sqrt( - c* 
*2*x**2 + 1)*a*c**4*x**4 + 161*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 63*sqr 
t( - c**2*x**2 + 1)*a + 693*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**12, 
x)*b*x**11 - 1386*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**10,x)*b*c**2* 
x**11 + 693*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**8,x)*b*c**4*x**11)) 
/(693*x**11)