\(\int (d+e x^2) (a+b \text {arccosh}(c x)) \, dx\) [465]

Optimal result

Integrand size = 16, antiderivative size = 94 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (9 c^2 d+2 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5908, 471, 75} \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+2 e\right )}{9 c^3}-\frac {b e x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c} \]

[In]

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

[Out]

-1/9*(b*(9*c^2*d + 2*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^3 - (b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) + d*x
*(a + b*ArcCosh[c*x]) + (e*x^3*(a + b*ArcCosh[c*x]))/3

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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 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 5908

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

Rubi steps \begin{align*} \text {integral}& = d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))-(b c) \int \frac {x \left (d+\frac {e x^2}{3}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))-\frac {1}{9} \left (b c \left (9 d+\frac {2 e}{c^2}\right )\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b \left (9 c^2 d+2 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+d x (a+b \text {arccosh}(c x))+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (9 d+e x^2\right )\right )}{c^3}+3 b x \left (3 d+e x^2\right ) \text {arccosh}(c x)\right ) \]

[In]

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

[Out]

(3*a*x*(3*d + e*x^2) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)))/c^3 + 3*b*x*(3*d + e*x^2)*Ar
cCosh[c*x])/9

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81

method	result	size
parts	\(a \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) x^{3} e}{3}+\operatorname {arccosh}\left (c x \right ) d c x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9 c^{2}}\right )}{c}\)	\(76\)
derivativedivides	\(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\)	\(90\)
default	\(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\)	\(90\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} e x^{2} + 9 \, b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c^{3}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{3} \, a e x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

int((a + b*acosh(c*x))*(d + e*x^2),x)

[Out]

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