\(\int x^4 (d+e x^2) (a+b \text {arccosh}(c x)) \, dx\) [360]

Optimal result

Integrand size = 19, antiderivative size = 177 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b \left (49 c^2 d+30 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^7}-\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x)) \] Output:

-8/3675*b*(49*c^2*d+30*e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^7-4/3675*b*(49*c^2 
*d+30*e)*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-1/1225*b*(49*c^2*d+30*e)*x^4* 
(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/49*b*e*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c 
+1/5*d*x^5*(a+b*arccosh(c*x))+1/7*e*x^7*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{35} a x^5 \left (7 d+5 e x^2\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e+8 c^2 \left (49 d+15 e x^2\right )+2 c^4 \left (98 d x^2+45 e x^4\right )+3 c^6 \left (49 d x^4+25 e x^6\right )\right )}{3675 c^7}+\frac {1}{35} b x^5 \left (7 d+5 e x^2\right ) \text {arccosh}(c x) \] Input:

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

Output:

(a*x^5*(7*d + 5*e*x^2))/35 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e + 8*c^ 
2*(49*d + 15*e*x^2) + 2*c^4*(98*d*x^2 + 45*e*x^4) + 3*c^6*(49*d*x^4 + 25*e 
*x^6)))/(3675*c^7) + (b*x^5*(7*d + 5*e*x^2)*ArcCosh[c*x])/35
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6371, 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.

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

\(\Big \downarrow \) 6371

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

\(\Big \downarrow \) 960

\(\displaystyle -\frac {1}{35} b c \left (\frac {1}{7} \left (\frac {30 e}{c^2}+49 d\right ) \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {5 e x^6 \sqrt {c x-1} \sqrt {c x+1}}{7 c^2}\right )+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{35} b c \left (\frac {1}{7} \left (\frac {30 e}{c^2}+49 d\right ) \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )+\frac {5 e x^6 \sqrt {c x-1} \sqrt {c x+1}}{7 c^2}\right )+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c \left (\frac {1}{7} \left (\frac {30 e}{c^2}+49 d\right ) \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )+\frac {5 e x^6 \sqrt {c x-1} \sqrt {c x+1}}{7 c^2}\right )+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{35} b c \left (\frac {1}{7} \left (\frac {30 e}{c^2}+49 d\right ) \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )+\frac {5 e x^6 \sqrt {c x-1} \sqrt {c x+1}}{7 c^2}\right )+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c \left (\frac {1}{7} \left (\frac {30 e}{c^2}+49 d\right ) \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )+\frac {5 e x^6 \sqrt {c x-1} \sqrt {c x+1}}{7 c^2}\right )+\frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 83

\(\displaystyle \frac {1}{5} d x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e x^7 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {5 e x^6 \sqrt {c x-1} \sqrt {c x+1}}{7 c^2}+\frac {1}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right ) \left (\frac {30 e}{c^2}+49 d\right )\right )\)

Input:

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

Output:

-1/35*(b*c*((5*e*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(7*c^2) + ((49*d + (30* 
e)/c^2)*((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 + (e*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 6371

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x 
_)^2), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*ArcCosh[c*x])/(f*(m + 1))) 
, x] + (Simp[e*(f*x)^(m + 3)*((a + b*ArcCosh[c*x])/(f^3*(m + 3))), x] - Sim 
p[b*(c/(f*(m + 1)*(m + 3)))   Int[(f*x)^(m + 1)*((d*(m + 3) + e*(m + 1)*x^2 
)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, 
x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69

method	result	size
parts	\(a \left (\frac {1}{7} e \,x^{7}+\frac {1}{5} d \,x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 x^{6} c^{6} e +147 x^{4} c^{6} d +90 x^{4} c^{4} e +196 x^{2} c^{4} d +120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right )}{3675 c^{2}}\right )}{c^{5}}\)	\(123\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 x^{6} c^{6} e +147 x^{4} c^{6} d +90 x^{4} c^{4} e +196 x^{2} c^{4} d +120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right )}{3675}\right )}{c^{2}}}{c^{5}}\)	\(133\)
default	\(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 x^{6} c^{6} e +147 x^{4} c^{6} d +90 x^{4} c^{4} e +196 x^{2} c^{4} d +120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right )}{3675}\right )}{c^{2}}}{c^{5}}\)	\(133\)
orering	\(\frac {\left (975 c^{8} e^{2} x^{10}+2442 c^{8} d e \,x^{8}+1323 c^{8} d^{2} x^{6}+90 x^{8} e^{2} c^{6}+354 x^{6} e \,c^{6} d +196 c^{6} d^{2} x^{4}+180 c^{4} e^{2} x^{6}+1296 c^{4} d e \,x^{4}+784 c^{4} d^{2} x^{2}+720 c^{2} e^{2} x^{4}-1872 c^{2} d e \,x^{2}-1568 c^{2} d^{2}-1440 e^{2} x^{2}-960 d e \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{3675 x \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (75 x^{6} c^{6} e +147 x^{4} c^{6} d +90 x^{4} c^{4} e +196 x^{2} c^{4} d +120 c^{2} e \,x^{2}+392 c^{2} d +240 e \right ) \left (c x -1\right ) \left (c x +1\right ) \left (4 x^{3} \left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+2 e \,x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+\frac {x^{4} \left (e \,x^{2}+d \right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3675 c^{8} \left (e \,x^{2}+d \right ) x^{4}}\)	\(309\)

Input:

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

Output:

a*(1/7*e*x^7+1/5*d*x^5)+b/c^5*(1/7*c^5*arccosh(c*x)*e*x^7+1/5*arccosh(c*x) 
*c^5*x^5*d-1/3675/c^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*e*x^6+147*c^6*d* 
x^4+90*c^4*e*x^4+196*c^4*d*x^2+120*c^2*e*x^2+392*c^2*d+240*e))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.79 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {525 \, a c^{7} e x^{7} + 735 \, a c^{7} d x^{5} + 105 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} e x^{6} + 3 \, {\left (49 \, b c^{6} d + 30 \, b c^{4} e\right )} x^{4} + 392 \, b c^{2} d + 4 \, {\left (49 \, b c^{4} d + 30 \, b c^{2} e\right )} x^{2} + 240 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{7}} \] Input:

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

Output:

1/3675*(525*a*c^7*e*x^7 + 735*a*c^7*d*x^5 + 105*(5*b*c^7*e*x^7 + 7*b*c^7*d 
*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*e*x^6 + 3*(49*b*c^6*d + 30* 
b*c^4*e)*x^4 + 392*b*c^2*d + 4*(49*b*c^4*d + 30*b*c^2*e)*x^2 + 240*b*e)*sq 
rt(c^2*x^2 - 1))/c^7
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

Integral(x**4*(a + b*acosh(c*x))*(d + e*x**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.01 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e \] Input:

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

Output:

1/7*a*e*x^7 + 1/5*a*d*x^5 + 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 
 + 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*e
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.05 \[ \int x^4 \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {735 \mathit {acosh} \left (c x \right ) b \,c^{7} d \,x^{5}+525 \mathit {acosh} \left (c x \right ) b \,c^{7} e \,x^{7}-147 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d \,x^{4}-75 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e \,x^{6}-196 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d \,x^{2}-90 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e \,x^{4}-392 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d -120 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e \,x^{2}-240 \sqrt {c^{2} x^{2}-1}\, b e +735 a \,c^{7} d \,x^{5}+525 a \,c^{7} e \,x^{7}}{3675 c^{7}} \] Input:

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

Output:

(735*acosh(c*x)*b*c**7*d*x**5 + 525*acosh(c*x)*b*c**7*e*x**7 - 147*sqrt(c* 
*2*x**2 - 1)*b*c**6*d*x**4 - 75*sqrt(c**2*x**2 - 1)*b*c**6*e*x**6 - 196*sq 
rt(c**2*x**2 - 1)*b*c**4*d*x**2 - 90*sqrt(c**2*x**2 - 1)*b*c**4*e*x**4 - 3 
92*sqrt(c**2*x**2 - 1)*b*c**2*d - 120*sqrt(c**2*x**2 - 1)*b*c**2*e*x**2 - 
240*sqrt(c**2*x**2 - 1)*b*e + 735*a*c**7*d*x**5 + 525*a*c**7*e*x**7)/(3675 
*c**7)