3.4.31 \(\int \frac {(1-c^2 x^2)^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx\) [331]

3.4.31.1 Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c x (a+b \text {arccosh}(c x))}-\frac {9 \sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 \sqrt {-1+c x}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Int}\left (\frac {-1+c^2 x^2}{x^2 (a+b \text {arccosh}(c x))},x\right )}{b c \sqrt {-1+c x}} \]

output

9/4*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-3 
/4*cosh(3*a/b)*Shi(3*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2 
)-9/4*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2) 
+3/4*Chi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1 
/2)-(-c^2*x^2+1)^(3/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/x/(a+b*arccosh(c*x) 
)-(-c*x+1)^(1/2)*Unintegrable((c^2*x^2-1)/x^2/(a+b*arccosh(c*x)),x)/b/c/(c 
*x-1)^(1/2)
 

3.4.31.2 Mathematica [N/A]

Not integrable

Time = 18.83 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx \]

input

Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])^2),x]
 

output

Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])^2), x]
 

3.4.31.3 Rubi [N/A]

Not integrable

Time = 1.67 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6357, 25, 6304, 6321, 25, 3042, 26, 3793, 2009, 6327, 6375}

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.

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

\(\Big \downarrow \) 6357

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6304

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

\(\Big \downarrow \) 6321

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-c x} \int \frac {(1-c x) (c x+1)}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {3 i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c x (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6327

\(\displaystyle \frac {\sqrt {1-c x} \int \frac {1-c^2 x^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {3 i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c x (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6375

\(\displaystyle \frac {\sqrt {1-c x} \int \frac {1-c^2 x^2}{x^2 (a+b \text {arccosh}(c x))}dx}{b c \sqrt {c x-1}}+\frac {3 i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c x (a+b \text {arccosh}(c x))}\)

input

Int[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])^2),x]
 

output

$Aborted
 

3.4.31.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6304

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + 
 d1*e2, 0] && IntegerQ[p]
 

rule 6321

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

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f*(n + 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] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

rule 6375

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
 

3.4.31.4 Maple [N/A] (verified)

Not integrable

Time = 1.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}d x\]

input

int((-c^2*x^2+1)^(3/2)/x/(a+b*arccosh(c*x))^2,x)
 

output

int((-c^2*x^2+1)^(3/2)/x/(a+b*arccosh(c*x))^2,x)
 

3.4.31.5 Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

input

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="fricas")
 

output

integral((-c^2*x^2 + 1)^(3/2)/(b^2*x*arccosh(c*x)^2 + 2*a*b*x*arccosh(c*x) 
 + a^2*x), x)
 

3.4.31.6 Sympy [N/A]

Not integrable

Time = 20.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

input

integrate((-c**2*x**2+1)**(3/2)/x/(a+b*acosh(c*x))**2,x)
 

output

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(x*(a + b*acosh(c*x))**2), x)
 

3.4.31.7 Maxima [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 476, normalized size of antiderivative = 17.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

input

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="maxima")
 

output

((c^4*x^4 - 2*c^2*x^2 + 1)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^5 - 2*c^3*x^3 
+ c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x 
 - 1)*a*b*c^2*x^2 - a*b*c*x + (b^2*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*b 
^2*c^2*x^2 - b^2*c*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate( 
((3*c^5*x^5 - c^3*x^3 - 2*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + (6*c^6*x^6 - 7* 
c^4*x^4 + 1)*(c*x + 1)*sqrt(c*x - 1) + 3*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*s 
qrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^5*x^6 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^ 
4 - 2*a*b*c^3*x^4 + a*b*c*x^2 + 2*(a*b*c^4*x^5 - a*b*c^2*x^3)*sqrt(c*x + 1 
)*sqrt(c*x - 1) + (b^2*c^5*x^6 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^4 - 2*b^2*c 
^3*x^4 + b^2*c*x^2 + 2*(b^2*c^4*x^5 - b^2*c^2*x^3)*sqrt(c*x + 1)*sqrt(c*x 
- 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

3.4.31.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arccosh(c*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
 

3.4.31.9 Mupad [N/A]

Not integrable

Time = 3.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

input

int((1 - c^2*x^2)^(3/2)/(x*(a + b*acosh(c*x))^2),x)
 

output

int((1 - c^2*x^2)^(3/2)/(x*(a + b*acosh(c*x))^2), x)