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

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}-\frac {b d \text {arccosh}(c x)}{24 c^4}+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x)) \] Output:

-1/24*b*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/36*b*d*x^3*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)/c+1/36*b*c*d*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/24*b*d*arccosh(c 
*x)/c^4+1/4*d*x^4*(a+b*arccosh(c*x))-1/6*c^2*d*x^6*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.23 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} a d x^4-\frac {1}{6} a c^2 d x^6-\frac {b d x \sqrt {-1+c x} \sqrt {1+c x}}{24 c^3}-\frac {b d x^3 \sqrt {-1+c x} \sqrt {1+c x}}{36 c}+\frac {1}{36} b c d x^5 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{4} b d x^4 \text {arccosh}(c x)-\frac {1}{6} b c^2 d x^6 \text {arccosh}(c x)-\frac {b d \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{12 c^4} \] Input:

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

Output:

(a*d*x^4)/4 - (a*c^2*d*x^6)/6 - (b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c 
^3) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) + (b*c*d*x^5*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x])/36 + (b*d*x^4*ArcCosh[c*x])/4 - (b*c^2*d*x^6*ArcCosh[ 
c*x])/6 - (b*d*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/(12*c^4)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6336, 27, 960, 111, 27, 101, 43}

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.

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

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int \frac {d x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{12} b c d \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 960

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 43

\(\displaystyle -\frac {1}{6} c^2 d x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c d \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\)

Input:

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

Output:

(d*x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*d*x^6*(a + b*ArcCosh[c*x]))/6 - (b*c 
*d*(-1/3*(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*((x^3*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x])/(4*c^2) + (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCos 
h[c*x]/(2*c^3)))/(4*c^2)))/3))/12
 

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 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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 = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.12

method	result	size
parts	\(-d a \left (\frac {1}{6} c^{2} x^{6}-\frac {1}{4} x^{4}\right )-\frac {d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-2 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+3 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{72 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(151\)
derivativedivides	\(\frac {-d a \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-2 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+3 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{72 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(155\)
default	\(\frac {-d a \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (-2 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+2 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+3 c x \sqrt {c^{2} x^{2}-1}+3 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{72 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(155\)
orering	\(\frac {\left (22 c^{6} x^{6}-34 c^{4} x^{4}-9 c^{2} x^{2}+12\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\left (2 c^{4} x^{4}-2 c^{2} x^{2}-3\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-2 x^{4} c^{2} d \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{72 x^{2} c^{4}}\)	\(162\)

Input:

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

Output:

-d*a*(1/6*c^2*x^6-1/4*x^4)-d*b/c^4*(1/6*arccosh(c*x)*c^6*x^6-1/4*arccosh(c 
*x)*c^4*x^4+1/72*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-2*c^5*x^5*(c^2*x^2-1)^(1/2) 
+2*(c^2*x^2-1)^(1/2)*c^3*x^3+3*c*x*(c^2*x^2-1)^(1/2)+3*ln(c*x+(c^2*x^2-1)^ 
(1/2)))/(c^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.80 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {12 \, a c^{6} d x^{6} - 18 \, a c^{4} d x^{4} + 3 \, {\left (4 \, b c^{6} d x^{6} - 6 \, b c^{4} d x^{4} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d x^{5} - 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt {c^{2} x^{2} - 1}}{72 \, c^{4}} \] Input:

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

Output:

-1/72*(12*a*c^6*d*x^6 - 18*a*c^4*d*x^4 + 3*(4*b*c^6*d*x^6 - 6*b*c^4*d*x^4 
+ b*d)*log(c*x + sqrt(c^2*x^2 - 1)) - (2*b*c^5*d*x^5 - 2*b*c^3*d*x^3 - 3*b 
*c*d*x)*sqrt(c^2*x^2 - 1))/c^4
 

Sympy [F]

\[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=- d \left (\int \left (- a x^{3}\right )\, dx + \int a c^{2} x^{5}\, dx + \int \left (- b x^{3} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{2} x^{5} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

-d*(Integral(-a*x**3, x) + Integral(a*c**2*x**5, x) + Integral(-b*x**3*aco 
sh(c*x), x) + Integral(b*c**2*x**5*acosh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.50 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{6} \, a c^{2} d x^{6} + \frac {1}{4} \, a d x^{4} - \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{2} d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d \] Input:

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

Output:

-1/6*a*c^2*d*x^6 + 1/4*a*d*x^4 - 1/288*(48*x^6*arccosh(c*x) - (8*sqrt(c^2* 
x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c 
^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*c^2*d + 1/32*(8*x^4 
*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 
 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*d
 

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-12 \mathit {acosh} \left (c x \right ) b \,c^{6} x^{6}+18 \mathit {acosh} \left (c x \right ) b \,c^{4} x^{4}+2 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} x^{5}-2 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3}-3 \sqrt {c^{2} x^{2}-1}\, b c x -3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b -12 a \,c^{6} x^{6}+18 a \,c^{4} x^{4}\right )}{72 c^{4}} \] Input:

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

Output:

(d*( - 12*acosh(c*x)*b*c**6*x**6 + 18*acosh(c*x)*b*c**4*x**4 + 2*sqrt(c**2 
*x**2 - 1)*b*c**5*x**5 - 2*sqrt(c**2*x**2 - 1)*b*c**3*x**3 - 3*sqrt(c**2*x 
**2 - 1)*b*c*x - 3*log(sqrt(c**2*x**2 - 1) + c*x)*b - 12*a*c**6*x**6 + 18* 
a*c**4*x**4))/(72*c**4)