\(\int x (d+e x^2) (a+b \text {arccosh}(c x)) \, dx\) [464]

Optimal result

Integrand size = 17, antiderivative size = 122 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b \left (8 c^2 d+3 e\right ) \text {arccosh}(c x)}{32 c^4}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x)) \]

[Out]

-1/32*b*(8*c^2*d+3*e)*arccosh(c*x)/c^4+1/2*d*x^2*(a+b*arccosh(c*x))+1/4*e*x^4*(a+b*arccosh(c*x))-1/32*b*(8*c^2
*d+3*e)*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/16*b*e*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5956, 471, 92, 54} \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {b \text {arccosh}(c x) \left (8 c^2 d+3 e\right )}{32 c^4}-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (8 c^2 d+3 e\right )}{32 c^3}-\frac {b e x^3 \sqrt {c x-1} \sqrt {c x+1}}{16 c} \]

[In]

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

[Out]

-1/32*(b*(8*c^2*d + 3*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^3 - (b*e*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) -
(b*(8*c^2*d + 3*e)*ArcCosh[c*x])/(32*c^4) + (d*x^2*(a + b*ArcCosh[c*x]))/2 + (e*x^4*(a + b*ArcCosh[c*x]))/4

Rule 54

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

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[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 471

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] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(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 5956

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] + (-Dist[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] + Simp[e*(f*x)^(m + 3)*((a + b*ArcCosh[c*x])/(f^3*(m
+ 3))), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {1}{8} (b c) \int \frac {x^2 \left (4 d+2 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {1}{16} \left (b c \left (8 d+\frac {3 e}{c^2}\right )\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x))-\frac {\left (b \left (8 c^2 d+3 e\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^3} \\ & = -\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b \left (8 c^2 d+3 e\right ) \text {arccosh}(c x)}{32 c^4}+\frac {1}{2} d x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e x^4 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {c x \left (8 a c^3 x \left (2 d+e x^2\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (3 e+2 c^2 \left (4 d+e x^2\right )\right )\right )+8 b c^4 x^2 \left (2 d+e x^2\right ) \text {arccosh}(c x)-2 b \left (8 c^2 d+3 e\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{32 c^4} \]

[In]

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

[Out]

(c*x*(8*a*c^3*x*(2*d + e*x^2) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3*e + 2*c^2*(4*d + e*x^2))) + 8*b*c^4*x^2*(2*d
+ e*x^2)*ArcCosh[c*x] - 2*b*(8*c^2*d + 3*e)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(32*c^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(224\) vs. \(2(104)=208\).

Time = 0.34 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.84

method	result	size
parts	\(\frac {a \left (e \,x^{2}+d \right )^{2}}{4 e}+\frac {b \left (\frac {c^{2} e \,\operatorname {arccosh}\left (c x \right ) x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{2} x^{2} d}{2}+\frac {c^{2} \operatorname {arccosh}\left (c x \right ) d^{2}}{4 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+8 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+8 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+3 e^{2} c x \sqrt {c^{2} x^{2}-1}+3 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 c^{2} e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(225\)
derivativedivides	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{2}}{4 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {e \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+8 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+8 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+3 e^{2} c x \sqrt {c^{2} x^{2}-1}+3 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c^{2}}\)	\(236\)
default	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{2}}{4 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {e \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+8 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+2 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+8 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+3 e^{2} c x \sqrt {c^{2} x^{2}-1}+3 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{32 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}}{c^{2}}\)	\(236\)

[In]

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

[Out]

1/4*a*(e*x^2+d)^2/e+b/c^2*(1/4*c^2*e*arccosh(c*x)*x^4+1/2*arccosh(c*x)*c^2*x^2*d+1/4*c^2/e*arccosh(c*x)*d^2-1/
32/c^2/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(8*c^4*d^2*ln(c*x+(c^2*x^2-1)^(1/2))+8*d*c^3*e*x*(c^2*x^2-1)^(1/2)+2*e^2*
(c^2*x^2-1)^(1/2)*c^3*x^3+8*d*c^2*e*ln(c*x+(c^2*x^2-1)^(1/2))+3*e^2*c*x*(c^2*x^2-1)^(1/2)+3*e^2*ln(c*x+(c^2*x^
2-1)^(1/2)))/(c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {8 \, a c^{4} e x^{4} + 16 \, a c^{4} d x^{2} + {\left (8 \, b c^{4} e x^{4} + 16 \, b c^{4} d x^{2} - 8 \, b c^{2} d - 3 \, b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{3} e x^{3} + {\left (8 \, b c^{3} d + 3 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, c^{4}} \]

[In]

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

[Out]

1/32*(8*a*c^4*e*x^4 + 16*a*c^4*d*x^2 + (8*b*c^4*e*x^4 + 16*b*c^4*d*x^2 - 8*b*c^2*d - 3*b*e)*log(c*x + sqrt(c^2
*x^2 - 1)) - (2*b*c^3*e*x^3 + (8*b*c^3*d + 3*b*c*e)*x)*sqrt(c^2*x^2 - 1))/c^4

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.28 \[ \int x \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b e \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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