\(\int \frac {(d+e x^2)^2 (a+b \text {arccosh}(c x))}{x^2} \, dx\) [375]

Optimal result

Integrand size = 21, antiderivative size = 141 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {2 b e \left (9 c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))+b c d^2 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:

-2/9*b*e*(9*c^2*d+e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/9*b*e^2*x^2*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)/c-d^2*(a+b*arccosh(c*x))/x+2*d*e*x*(a+b*arccosh(c*x)) 
+1/3*e^2*x^3*(a+b*arccosh(c*x))+b*c*d^2*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2) 
)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} \left (-\frac {3 a d^2}{x}+6 a d e x+a e^2 x^3-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (18 d+e x^2\right )\right )}{3 c^3}+\frac {b \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \text {arccosh}(c x)}{x}-3 b c d^2 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right ) \] Input:

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

Output:

((-3*a*d^2)/x + 6*a*d*e*x + a*e^2*x^3 - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]* 
(2*e + c^2*(18*d + e*x^2)))/(3*c^3) + (b*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*Ar 
cCosh[c*x])/x - 3*b*c*d^2*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])])/3
 

Rubi [A] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6373, 27, 1905, 1578, 1192, 1467, 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.

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

\(\Big \downarrow \) 6373

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1192

\(\displaystyle \frac {b \sqrt {c^2 x^2-1} \int \frac {-e^2 x^8-2 e \left (3 d c^2+e\right ) x^4+3 c^4 d^2-e^2-6 c^2 d e}{x^4+1}d\sqrt {c^2 x^2-1}}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}+2 d e x (a+b \text {arccosh}(c x))+\frac {1}{3} e^2 x^3 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1467

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d^2*(a + b*ArcCosh[c*x]))/x) + 2*d*e*x*(a + b*ArcCosh[c*x]) + (e^2*x^3* 
(a + b*ArcCosh[c*x]))/3 + (b*Sqrt[-1 + c^2*x^2]*(-1/3*(e^2*x^6) - e*(6*c^2 
*d + e)*Sqrt[-1 + c^2*x^2] + 3*c^4*d^2*ArcTan[Sqrt[-1 + c^2*x^2]]))/(3*c^3 
*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6373

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]}, Sim 
p[(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] && 
NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le 
Q[m + p, 0]))
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.26

method	result	size
parts	\(a \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+b c \left (\frac {\operatorname {arccosh}\left (c x \right ) e^{2} x^{3}}{3 c}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) x d e}{c}-\frac {\operatorname {arccosh}\left (c x \right ) d^{2}}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 d^{2} c^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+18 d \,c^{2} e \sqrt {c^{2} x^{2}-1}+e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\right )}{9 c^{4} \sqrt {c^{2} x^{2}-1}}\right )\)	\(178\)
derivativedivides	\(c \left (\frac {a \left (2 d \,c^{3} e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {d^{2} c^{3}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{3} e x +\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{3}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 d^{2} c^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+18 d \,c^{2} e \sqrt {c^{2} x^{2}-1}+e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\right )\)	\(191\)
default	\(c \left (\frac {a \left (2 d \,c^{3} e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {d^{2} c^{3}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{3} e x +\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{3}}{x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 d^{2} c^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+18 d \,c^{2} e \sqrt {c^{2} x^{2}-1}+e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\right )\)	\(191\)

Input:

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

Output:

a*(1/3*e^2*x^3+2*d*e*x-d^2/x)+b*c*(1/3/c*arccosh(c*x)*e^2*x^3+2/c*arccosh( 
c*x)*x*d*e-arccosh(c*x)*d^2/c/x-1/9/c^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(9*d^2 
*c^4*arctan(1/(c^2*x^2-1)^(1/2))+18*d*c^2*e*(c^2*x^2-1)^(1/2)+e^2*c^2*x^2* 
(c^2*x^2-1)^(1/2)+2*e^2*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.67 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {3 \, a c^{3} e^{2} x^{4} + 18 \, b c^{4} d^{2} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 18 \, a c^{3} d e x^{2} - 9 \, a c^{3} d^{2} + 3 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 3 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} e^{2} x^{3} + 2 \, {\left (9 \, b c^{2} d e + b e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c^{3} x} \] Input:

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

Output:

1/9*(3*a*c^3*e^2*x^4 + 18*b*c^4*d^2*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 1 
8*a*c^3*d*e*x^2 - 9*a*c^3*d^2 + 3*(3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)* 
x*log(-c*x + sqrt(c^2*x^2 - 1)) + 3*(b*c^3*e^2*x^4 + 6*b*c^3*d*e*x^2 - 3*b 
*c^3*d^2 + (3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x)*log(c*x + sqrt(c^2*x 
^2 - 1)) - (b*c^2*e^2*x^3 + 2*(9*b*c^2*d*e + b*e^2)*x)*sqrt(c^2*x^2 - 1))/ 
(c^3*x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} \, a e^{2} x^{3} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e^{2} + 2 \, a d e x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d e}{c} - \frac {a d^{2}}{x} \] Input:

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

Output:

1/3*a*e^2*x^3 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b*d^2 + 1/9*(3*x 
^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)) 
*b*e^2 + 2*a*d*e*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d*e/c - a* 
d^2/x
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {-9 \mathit {acosh} \left (c x \right ) b \,c^{3} d^{2}+18 \mathit {acosh} \left (c x \right ) b \,c^{3} d e \,x^{2}+3 \mathit {acosh} \left (c x \right ) b \,c^{3} e^{2} x^{4}-18 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{4} d^{2} x -\sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{2} x^{3}-2 \sqrt {c^{2} x^{2}-1}\, b \,e^{2} x -18 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{2} d e x -9 a \,c^{3} d^{2}+18 a \,c^{3} d e \,x^{2}+3 a \,c^{3} e^{2} x^{4}}{9 c^{3} x} \] Input:

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

Output:

( - 9*acosh(c*x)*b*c**3*d**2 + 18*acosh(c*x)*b*c**3*d*e*x**2 + 3*acosh(c*x 
)*b*c**3*e**2*x**4 - 18*atan(sqrt(c**2*x**2 - 1) + c*x)*b*c**4*d**2*x - sq 
rt(c**2*x**2 - 1)*b*c**2*e**2*x**3 - 2*sqrt(c**2*x**2 - 1)*b*e**2*x - 18*s 
qrt(c*x + 1)*sqrt(c*x - 1)*b*c**2*d*e*x - 9*a*c**3*d**2 + 18*a*c**3*d*e*x* 
*2 + 3*a*c**3*e**2*x**4)/(9*c**3*x)