\(\int \frac {(d-c^2 d x^2)^2}{a+b \text {arccosh}(c x)} \, dx\) [29]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{a+b \text {arccosh}(c x)} \, dx=-\frac {5 d^2 \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b c}+\frac {5 d^2 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b c}-\frac {d^2 \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b c}+\frac {5 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c}-\frac {5 d^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c}+\frac {d^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c} \] Output:

-5/8*d^2*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)/b/c+5/16*d^2*Chi(3*(a+b*arcco 
sh(c*x))/b)*sinh(3*a/b)/b/c-1/16*d^2*Chi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/ 
b)/b/c+5/8*d^2*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)/b/c-5/16*d^2*cosh(3*a/b 
)*Shi(3*(a+b*arccosh(c*x))/b)/b/c+1/16*d^2*cosh(5*a/b)*Shi(5*(a+b*arccosh( 
c*x))/b)/b/c
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{a+b \text {arccosh}(c x)} \, dx=-\frac {d^2 \left (10 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-5 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-10 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b c} \] Input:

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

Output:

-1/16*(d^2*(10*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 5*CoshIntegral 
[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] + CoshIntegral[5*(a/b + ArcCosh[c*x 
])]*Sinh[(5*a)/b] - 10*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 5*Cosh 
[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - Cosh[(5*a)/b]*SinhIntegra 
l[5*(a/b + ArcCosh[c*x])]))/(b*c)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6321, 25, 3042, 26, 3793, 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.

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

\(\Big \downarrow \) 6321

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d^2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^5}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i d^2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^5}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {i d^2 \int \left (\frac {i \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {5 i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}+\frac {5 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i d^2 \left (\frac {5}{8} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {5}{16} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{8} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {5}{16} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c}\)

Input:

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

Output:

(I*d^2*(((5*I)/8)*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b] - ((5*I)/ 
16)*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b] + (I/16)*CoshIn 
tegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b] - ((5*I)/8)*Cosh[a/b]*Sin 
hIntegral[(a + b*ArcCosh[c*x])/b] + ((5*I)/16)*Cosh[(3*a)/b]*SinhIntegral[ 
(3*(a + b*ArcCosh[c*x]))/b] - (I/16)*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b* 
ArcCosh[c*x]))/b]))/(b*c)
 

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 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]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{32 b}-\frac {5 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{32 b}+\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{16 b}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{16 b}+\frac {5 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{32 b}-\frac {d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{32 b}}{c}\)	\(182\)
default	\(\frac {\frac {d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{32 b}-\frac {5 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{32 b}+\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{16 b}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{16 b}+\frac {5 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{32 b}-\frac {d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{32 b}}{c}\)	\(182\)

Input:

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

Output:

1/c*(1/32*d^2/b*exp(5*a/b)*Ei(1,5*arccosh(c*x)+5*a/b)-5/32*d^2/b*exp(3*a/b 
)*Ei(1,3*arccosh(c*x)+3*a/b)+5/16*d^2/b*exp(a/b)*Ei(1,arccosh(c*x)+a/b)-5/ 
16*d^2/b*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)+5/32*d^2/b*exp(-3*a/b)*Ei(1,-3* 
arccosh(c*x)-3*a/b)-1/32*d^2/b*exp(-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

d**2*(Integral(-2*c**2*x**2/(a + b*acosh(c*x)), x) + Integral(c**4*x**4/(a 
 + b*acosh(c*x)), x) + Integral(1/(a + b*acosh(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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