\(\int \frac {(d-c^2 d x^2)^3}{a+b \text {arccosh}(c x)} \, dx\) [28]

Optimal result

Integrand size = 24, antiderivative size = 269 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \text {arccosh}(c x)} \, dx=-\frac {35 d^3 \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b c}+\frac {21 d^3 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b c}-\frac {7 d^3 \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b c}+\frac {d^3 \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b c}+\frac {35 d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b c}-\frac {21 d^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c}+\frac {7 d^3 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c}-\frac {d^3 \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b c} \] Output:

-35/64*d^3*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)/b/c+21/64*d^3*Chi(3*(a+b*ar 
ccosh(c*x))/b)*sinh(3*a/b)/b/c-7/64*d^3*Chi(5*(a+b*arccosh(c*x))/b)*sinh(5 
*a/b)/b/c+1/64*d^3*Chi(7*(a+b*arccosh(c*x))/b)*sinh(7*a/b)/b/c+35/64*d^3*c 
osh(a/b)*Shi((a+b*arccosh(c*x))/b)/b/c-21/64*d^3*cosh(3*a/b)*Shi(3*(a+b*ar 
ccosh(c*x))/b)/b/c+7/64*d^3*cosh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)/b/c-1/ 
64*d^3*cosh(7*a/b)*Shi(7*(a+b*arccosh(c*x))/b)/b/c
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{a+b \text {arccosh}(c x)} \, dx=\frac {d^3 \left (-35 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+21 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-7 \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+\text {Chi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )+35 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-21 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+7 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{64 b c} \] Input:

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

Output:

(d^3*(-35*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 21*CoshIntegral[3*( 
a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 7*CoshIntegral[5*(a/b + ArcCosh[c*x]) 
]*Sinh[(5*a)/b] + CoshIntegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 35* 
Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 21*Cosh[(3*a)/b]*SinhIntegral 
[3*(a/b + ArcCosh[c*x])] + 7*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c 
*x])] - Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))/(64*b*c)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.84, 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.

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

\(\Big \downarrow \) 6321

\(\displaystyle -\frac {d^3 \int -\frac {\sinh ^7\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^3 \int \frac {\sinh ^7\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^3 \int \frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^7}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i d^3 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^7}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {i d^3 \int \left (-\frac {i \sinh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}+\frac {7 i \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}-\frac {21 i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}+\frac {35 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i d^3 \left (\frac {35}{64} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {21}{64} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {7}{64} i \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{64} i \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )-\frac {35}{64} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {21}{64} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {7}{64} i \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{64} i \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c}\)

Input:

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

Output:

(I*d^3*(((35*I)/64)*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b] - ((21* 
I)/64)*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b] + ((7*I)/64) 
*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b] - (I/64)*CoshInteg 
ral[(7*(a + b*ArcCosh[c*x]))/b]*Sinh[(7*a)/b] - ((35*I)/64)*Cosh[a/b]*Sinh 
Integral[(a + b*ArcCosh[c*x])/b] + ((21*I)/64)*Cosh[(3*a)/b]*SinhIntegral[ 
(3*(a + b*ArcCosh[c*x]))/b] - ((7*I)/64)*Cosh[(5*a)/b]*SinhIntegral[(5*(a 
+ b*ArcCosh[c*x]))/b] + (I/64)*Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCos 
h[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 = 0.45 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {-\frac {d^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right )}{128 b}+\frac {7 d^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{128 b}-\frac {21 d^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{128 b}+\frac {35 d^{3} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{128 b}-\frac {35 d^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{128 b}+\frac {21 d^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{128 b}-\frac {7 d^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{128 b}+\frac {d^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right )}{128 b}}{c}\)	\(242\)
default	\(\frac {-\frac {d^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right )}{128 b}+\frac {7 d^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{128 b}-\frac {21 d^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{128 b}+\frac {35 d^{3} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{128 b}-\frac {35 d^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{128 b}+\frac {21 d^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{128 b}-\frac {7 d^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{128 b}+\frac {d^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right )}{128 b}}{c}\)	\(242\)

Input:

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

Output:

1/c*(-1/128*d^3/b*exp(7*a/b)*Ei(1,7*arccosh(c*x)+7*a/b)+7/128*d^3/b*exp(5* 
a/b)*Ei(1,5*arccosh(c*x)+5*a/b)-21/128*d^3/b*exp(3*a/b)*Ei(1,3*arccosh(c*x 
)+3*a/b)+35/128*d^3/b*exp(a/b)*Ei(1,arccosh(c*x)+a/b)-35/128*d^3/b*exp(-a/ 
b)*Ei(1,-arccosh(c*x)-a/b)+21/128*d^3/b*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3 
*a/b)-7/128*d^3/b*exp(-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b)+1/128*d^3/b*exp( 
-7*a/b)*Ei(1,-7*arccosh(c*x)-7*a/b))
 

Fricas [F]

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

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

Output:

integral(-(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)/(b*arccosh(c 
*x) + a), x)
 

Sympy [F]

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

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

Output:

-d**3*(Integral(3*c**2*x**2/(a + b*acosh(c*x)), x) + Integral(-3*c**4*x**4 
/(a + b*acosh(c*x)), x) + Integral(c**6*x**6/(a + b*acosh(c*x)), x) + Inte 
gral(-1/(a + b*acosh(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d**3*( - int(x**6/(acosh(c*x)*b + a),x)*c**6 + 3*int(x**4/(acosh(c*x)*b + 
a),x)*c**4 - 3*int(x**2/(acosh(c*x)*b + a),x)*c**2 + int(1/(acosh(c*x)*b + 
 a),x))