\(\int x^4 (d-c^2 d x^2) (a+b \text {arccosh}(c x)) \, dx\) [1]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {152 b d \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {76 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^3}-\frac {19 b d x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c}+\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x)) \] Output:

-152/3675*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-76/3675*b*d*x^2*(c*x-1)^(1/2 
)*(c*x+1)^(1/2)/c^3-19/1225*b*d*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+1/49*b*c 
*d*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/5*d*x^5*(a+b*arccosh(c*x))-1/7*c^2*d* 
x^7*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.60 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-105 a x^5 \left (-7+5 c^2 x^2\right )+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (-152-76 c^2 x^2-57 c^4 x^4+75 c^6 x^6\right )}{c^5}-105 b x^5 \left (-7+5 c^2 x^2\right ) \text {arccosh}(c x)\right )}{3675} \] Input:

Integrate[x^4*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]
 

Output:

(d*(-105*a*x^5*(-7 + 5*c^2*x^2) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-152 - 
76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6))/c^5 - 105*b*x^5*(-7 + 5*c^2*x^2)*Ar 
cCosh[c*x]))/3675
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6336, 27, 960, 111, 27, 111, 27, 83}

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.

\(\displaystyle \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int \frac {d x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c d \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 960

\(\displaystyle -\frac {1}{35} b c d \left (\frac {19}{7} \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{35} b c d \left (\frac {19}{7} \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c d \left (\frac {19}{7} \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{35} b c d \left (\frac {19}{7} \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c d \left (\frac {19}{7} \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 83

\(\displaystyle -\frac {1}{7} c^2 d x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c d \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\)

Input:

Int[x^4*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]
 

Output:

-1/35*(b*c*d*((-5*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/7 + (19*((x^4*Sqrt[-1 
+ c*x]*Sqrt[1 + c*x])/(5*c^2) + (4*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^ 
4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2)))/(5*c^2)))/7)) + (d*x^5*( 
a + b*ArcCosh[c*x]))/5 - (c^2*d*x^7*(a + b*ArcCosh[c*x]))/7
 

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 83

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f 
*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 6336

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.62

method	result	size
parts	\(-d a \left (\frac {1}{7} c^{2} x^{7}-\frac {1}{5} x^{5}\right )-\frac {d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-57 c^{4} x^{4}-76 c^{2} x^{2}-152\right )}{3675}\right )}{c^{5}}\)	\(94\)
derivativedivides	\(\frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-57 c^{4} x^{4}-76 c^{2} x^{2}-152\right )}{3675}\right )}{c^{5}}\)	\(98\)
default	\(\frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-57 c^{4} x^{4}-76 c^{2} x^{2}-152\right )}{3675}\right )}{c^{5}}\)	\(98\)
orering	\(\frac {\left (975 c^{8} x^{8}-1377 c^{6} x^{6}-228 c^{4} x^{4}-608 c^{2} x^{2}+608\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{3675 c^{6} x \left (c^{2} x^{2}-1\right )}-\frac {\left (75 c^{6} x^{6}-57 c^{4} x^{4}-76 c^{2} x^{2}-152\right ) \left (4 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-2 c^{2} d \,x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3675 c^{6} x^{4}}\)	\(181\)

Input:

int(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

-d*a*(1/7*c^2*x^7-1/5*x^5)-d*b/c^5*(1/7*arccosh(c*x)*c^7*x^7-1/5*arccosh(c 
*x)*c^5*x^5-1/3675*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*x^6-57*c^4*x^4-76*c 
^2*x^2-152))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {525 \, a c^{7} d x^{7} - 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} - 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} d x^{6} - 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} - 152 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{5}} \] Input:

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")
 

Output:

-1/3675*(525*a*c^7*d*x^7 - 735*a*c^5*d*x^5 + 105*(5*b*c^7*d*x^7 - 7*b*c^5* 
d*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*d*x^6 - 57*b*c^4*d*x^4 - 7 
6*b*c^2*d*x^2 - 152*b*d)*sqrt(c^2*x^2 - 1))/c^5
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(x**4*(-c**2*d*x**2+d)*(a+b*acosh(c*x)),x)
 

Output:

-d*(Integral(-a*x**4, x) + Integral(a*c**2*x**6, x) + Integral(-b*x**4*aco 
sh(c*x), x) + Integral(b*c**2*x**6*acosh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d \] Input:

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")
 

Output:

-1/7*a*c^2*d*x^7 + 1/5*a*d*x^5 - 1/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2* 
x^2 - 1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c 
^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^2*d + 1/75*(15*x^5*arccosh(c*x) - (3 
*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 
- 1)/c^6)*c)*b*d
 

Giac [F(-2)]

Exception generated. \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-525 \mathit {acosh} \left (c x \right ) b \,c^{7} x^{7}+735 \mathit {acosh} \left (c x \right ) b \,c^{5} x^{5}+75 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} x^{6}-57 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} x^{4}-76 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} x^{2}-152 \sqrt {c^{2} x^{2}-1}\, b -525 a \,c^{7} x^{7}+735 a \,c^{5} x^{5}\right )}{3675 c^{5}} \] Input:

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

Output:

(d*( - 525*acosh(c*x)*b*c**7*x**7 + 735*acosh(c*x)*b*c**5*x**5 + 75*sqrt(c 
**2*x**2 - 1)*b*c**6*x**6 - 57*sqrt(c**2*x**2 - 1)*b*c**4*x**4 - 76*sqrt(c 
**2*x**2 - 1)*b*c**2*x**2 - 152*sqrt(c**2*x**2 - 1)*b - 525*a*c**7*x**7 + 
735*a*c**5*x**5))/(3675*c**5)