\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx\) [76]

Optimal result

Integrand size = 27, antiderivative size = 409 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/110*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^10/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/66*b* 
c^3*d*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/1386*b*c^5*d* 
(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/770*b*c^7*d*(-c^2*d 
*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/1155*b*c^9*d*(-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)^(5/2)*(a+b*arc 
cosh(c*x))/d/x^11-2/33*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^9-8 
/231*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^7-16/1155*c^6*(-c^2*d 
*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^5+16/1155*b*c^11*d*(-c^2*d*x^2+d)^(1/ 
2)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.42 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (630 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+12 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (35+20 c^2 x^2+8 c^4 x^4\right ) (a+b \text {arccosh}(c x))+b c x \left (63-105 c^2 x^2+5 c^4 x^4+9 c^6 x^6+24 c^8 x^8-96 c^{10} x^{10} \log (x)\right )\right )}{6930 x^{11} \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]
 

Output:

-1/6930*(d*Sqrt[d - c^2*d*x^2]*(630*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + 
b*ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(35 + 20*c^2 
*x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]) + b*c*x*(63 - 105*c^2*x^2 + 5*c^4*x 
^4 + 9*c^6*x^6 + 24*c^8*x^8 - 96*c^10*x^10*Log[x])))/(x^11*Sqrt[-1 + c*x]* 
Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 2331, 2123, 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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx\)

\(\Big \downarrow \) 6337

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{1155 x^{11}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{x^{11}}dx}{1155 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{x^{12}}dx^2}{2310 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\)

\(\Big \downarrow \) 2123

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {16 c^{10}}{x^2}+\frac {8 c^8}{x^4}+\frac {6 c^6}{x^6}+\frac {5 c^4}{x^8}-\frac {140 c^2}{x^{10}}+\frac {105}{x^{12}}\right )dx^2}{2310 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}+\frac {b c d \sqrt {d-c^2 d x^2} \left (16 c^{10} \log \left (x^2\right )-\frac {8 c^8}{x^2}-\frac {3 c^6}{x^4}-\frac {5 c^4}{3 x^6}+\frac {35 c^2}{x^8}-\frac {21}{x^{10}}\right )}{2310 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]
 

Output:

-1/11*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(d*x^11) - (2*c^2*(d - 
c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2) 
^(5/2)*(a + b*ArcCosh[c*x]))/(231*d*x^7) - (16*c^6*(d - c^2*d*x^2)^(5/2)*( 
a + b*ArcCosh[c*x]))/(1155*d*x^5) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-21/x^10 + 
 (35*c^2)/x^8 - (5*c^4)/(3*x^6) - (3*c^6)/x^4 - (8*c^8)/x^2 + 16*c^10*Log[ 
x^2]))/(2310*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 2009

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

rule 2123

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

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

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. \(5522\) vs. \(2(345)=690\).

Time = 0.62 (sec) , antiderivative size = 5523, normalized size of antiderivative = 13.50

method	result	size
default	\(\text {Expression too large to display}\)	\(5523\)
parts	\(\text {Expression too large to display}\)	\(5523\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^12,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.94 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx =\text {Too large to display} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^12,x, algorithm="frica 
s")
 

Output:

[-1/6930*(6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b*c^6*d* 
x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*lo 
g(c*x + sqrt(c^2*x^2 - 1)) - 48*(b*c^13*d*x^13 - b*c^11*d*x^11)*sqrt(-d)*l 
og((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)) + (24*b*c^9*d*x^9 + 9*b*c^7*d*x^ 
7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*d 
*x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1 
) + 6*(16*a*c^12*d*x^12 - 8*a*c^10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 
145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^ 
13 - x^11), 1/6930*(96*(b*c^13*d*x^13 - b*c^11*d*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)) - 6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b* 
c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + 
 d)*log(c*x + sqrt(c^2*x^2 - 1)) - (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b 
*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*d*x^5 - 10 
5*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*(16 
*a*c^12*d*x^12 - 8*a*c^10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4 
*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11 
)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**12,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\frac {1}{6930} \, {\left (96 \, c^{10} \sqrt {-d} d \log \left (x\right ) - \frac {24 \, c^{8} \sqrt {-d} d x^{8} + 9 \, c^{6} \sqrt {-d} d x^{6} + 5 \, c^{4} \sqrt {-d} d x^{4} - 105 \, c^{2} \sqrt {-d} d x^{2} + 63 \, \sqrt {-d} d}{x^{10}}\right )} b c - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} a \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^12,x, algorithm="maxim 
a")
 

Output:

1/6930*(96*c^10*sqrt(-d)*d*log(x) - (24*c^8*sqrt(-d)*d*x^8 + 9*c^6*sqrt(-d 
)*d*x^6 + 5*c^4*sqrt(-d)*d*x^4 - 105*c^2*sqrt(-d)*d*x^2 + 63*sqrt(-d)*d)/x 
^10)*b*c - 1/1155*(16*(-c^2*d*x^2 + d)^(5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 
+ d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^9) + 105*(-c^2 
*d*x^2 + d)^(5/2)/(d*x^11))*b*arccosh(c*x) - 1/1155*(16*(-c^2*d*x^2 + d)^( 
5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*x^2 
+ d)^(5/2)*c^2/(d*x^9) + 105*(-c^2*d*x^2 + d)^(5/2)/(d*x^11))*a
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{12}} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^12,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^12, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\frac {\sqrt {d}\, d \left (-16 \sqrt {-c^{2} x^{2}+1}\, a \,c^{10} x^{10}-8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}-6 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-5 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+140 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-105 \sqrt {-c^{2} x^{2}+1}\, a +1155 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{12}}d x \right ) b \,x^{11}-1155 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{10}}d x \right ) b \,c^{2} x^{11}\right )}{1155 x^{11}} \] Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))/x^12,x)
 

Output:

(sqrt(d)*d*( - 16*sqrt( - c**2*x**2 + 1)*a*c**10*x**10 - 8*sqrt( - c**2*x* 
*2 + 1)*a*c**8*x**8 - 6*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 5*sqrt( - c** 
2*x**2 + 1)*a*c**4*x**4 + 140*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 105*sqr 
t( - c**2*x**2 + 1)*a + 1155*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**12 
,x)*b*x**11 - 1155*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**10,x)*b*c**2 
*x**11))/(1155*x**11)