\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx\) [73]

Optimal result

Integrand size = 27, antiderivative size = 166 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{5 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 d x^5}+\frac {b c^5 d \sqrt {d-c^2 d x^2} \log (x)}{5 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/20*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/5*b*c^3 
*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/5*(-c^2*d*x^2+d) 
^(5/2)*(a+b*arccosh(c*x))/d/x^5+1/5*b*c^5*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 = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.57 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (\frac {(-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))}{x^5}-b c \left (-\frac {1}{4 x^4}+\frac {c^2}{x^2}+c^4 \log (x)\right )\right )}{5 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^6,x]
 

Output:

-1/5*(d*Sqrt[d - c^2*d*x^2]*(((-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcC 
osh[c*x]))/x^5 - b*c*(-1/4*1/x^4 + c^2/x^2 + c^4*Log[x])))/(Sqrt[-1 + c*x] 
*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6332, 82, 243, 49, 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.

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

\(\Big \downarrow \) 6332

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^6,x]
 

Output:

-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(d*x^5) + (b*c*d*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])
 

Defintions of rubi rules used

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 2009

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

rule 6332

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 + 1)*((a + 
 b*ArcCosh[c*x])^n/(d*f*(m + 1))), 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, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 
, 0] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2170\) vs. \(2(138)=276\).

Time = 0.45 (sec) , antiderivative size = 2171, normalized size of antiderivative = 13.08

method	result	size
default	\(\text {Expression too large to display}\)	\(2171\)
parts	\(\text {Expression too large to display}\)	\(2171\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^6,x,method=_RETURNVERBOSE)
 

Output:

-11*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+ 
1)*x^5/(c*x-1)/(c*x+1)*arccosh(c*x)*c^10+14*b*(-d*(c^2*x^2-1))^(1/2)*d/(5* 
c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^3/(c*x-1)/(c*x+1)*arccosh(c*x 
)*c^8-56/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c 
^2*x^2+1)*x/(c*x-1)/(c*x+1)*arccosh(c*x)*c^6+28/5*b*(-d*(c^2*x^2-1))^(1/2) 
*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/x/(c*x-1)/(c*x+1)*arccosh 
(c*x)*c^4-8/5*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4- 
5*c^2*x^2+1)/x^3/(c*x-1)/(c*x+1)*arccosh(c*x)*c^2-b*(-d*(c^2*x^2-1))^(1/2) 
*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^9/(c*x-1)/(c*x+1)*arcco 
sh(c*x)*c^14-b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5 
*c^2*x^2+1)*x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^7+b*(-d*(c^2*x^ 
2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^8/(c*x-1)^(1 
/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^13-2*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^ 
8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arcco 
sh(c*x)*c^11+2*b*(-d*(c^2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4 
-5*c^2*x^2+1)*x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^9+5*b*(-d*(c^ 
2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^7/(c*x-1 
)/(c*x+1)*arccosh(c*x)*c^12-1/5*a/d/x^5*(-c^2*d*x^2+d)^(5/2)-3/2*b*(-d*(c^ 
2*x^2-1))^(1/2)*d/(5*c^8*x^8-10*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/(c*x-1)^(1 
/2)/(c*x+1)^(1/2)*c^5+1/5*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.45 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\left [-\frac {4 \, {\left (b c^{6} d x^{6} - 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} - b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (b c^{7} d x^{7} - b c^{5} d x^{5}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) - {\left (4 \, b c^{3} d x^{3} - {\left (4 \, b c^{3} - b c\right )} d x^{5} - b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 4 \, {\left (a c^{6} d x^{6} - 3 \, a c^{4} d x^{4} + 3 \, a c^{2} d x^{2} - a d\right )} \sqrt {-c^{2} d x^{2} + d}}{20 \, {\left (c^{2} x^{7} - x^{5}\right )}}, \frac {4 \, {\left (b c^{7} d x^{7} - b c^{5} d x^{5}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} - 1\right )} \sqrt {d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) - 4 \, {\left (b c^{6} d x^{6} - 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} - b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (4 \, b c^{3} d x^{3} - {\left (4 \, b c^{3} - b c\right )} d x^{5} - b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (a c^{6} d x^{6} - 3 \, a c^{4} d x^{4} + 3 \, a c^{2} d x^{2} - a d\right )} \sqrt {-c^{2} d x^{2} + d}}{20 \, {\left (c^{2} x^{7} - x^{5}\right )}}\right ] \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="fricas 
")
 

Output:

[-1/20*(4*(b*c^6*d*x^6 - 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 - b*d)*sqrt(-c^2*d* 
x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(b*c^7*d*x^7 - b*c^5*d*x^5)*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)) - (4*b*c^3*d*x^3 - (4*b*c^ 
3 - b*c)*d*x^5 - b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 4*(a*c^ 
6*d*x^6 - 3*a*c^4*d*x^4 + 3*a*c^2*d*x^2 - a*d)*sqrt(-c^2*d*x^2 + d))/(c^2* 
x^7 - x^5), 1/20*(4*(b*c^7*d*x^7 - b*c^5*d*x^5)*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)) - 4*(b*c^6*d*x^6 - 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 - b*d)*sqrt(-c^2*d 
*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (4*b*c^3*d*x^3 - (4*b*c^3 - b*c)* 
d*x^5 - b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 4*(a*c^6*d*x^6 - 
 3*a*c^4*d*x^4 + 3*a*c^2*d*x^2 - a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^7 - x^5 
)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**6,x)
 

Output:

Timed out
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.14 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {{\left (2 \, c^{6} d^{3} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) + 2 i \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{4} d^{\frac {5}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + \frac {3 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} c^{2} d^{2}}{x^{2}} - \frac {\sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} d^{2}}{x^{4}}\right )} b c}{20 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b \operatorname {arcosh}\left (c x\right )}{5 \, d x^{5}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a}{5 \, d x^{5}} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="maxima 
")
 

Output:

-1/20*(2*c^6*d^3*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + 2*I*(-1)^(-2*c^2*d*x^ 
2 + 2*d)*c^4*d^(5/2)*log(-2*c^2*d + 2*d/x^2) + 3*sqrt(-c^4*d*x^4 + 2*c^2*d 
*x^2 - d)*c^2*d^2/x^2 - sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*d^2/x^4)*b*c/d 
- 1/5*(-c^2*d*x^2 + d)^(5/2)*b*arccosh(c*x)/(d*x^5) - 1/5*(-c^2*d*x^2 + d) 
^(5/2)*a/(d*x^5)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^6} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^6,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^6, x)
 

Reduce [F]

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

int((-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))/x^6,x)
 

Output:

(sqrt(d)*d*( - sqrt( - c**2*x**2 + 1)*a*c**4*x**4 + 2*sqrt( - c**2*x**2 + 
1)*a*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a + 5*int((sqrt( - c**2*x**2 + 1)* 
acosh(c*x))/x**6,x)*b*x**5 - 5*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x** 
4,x)*b*c**2*x**5))/(5*x**5)