\(\int (d-c^2 d x^2) (a+b \text {arccosh}(c x)) \, dx\) [5]

Optimal result

Integrand size = 20, antiderivative size = 86 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+\frac {1}{9} b c d x^2 \sqrt {-1+c x} \sqrt {1+c x}+d x (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d x^3 (a+b \text {arccosh}(c x)) \]

[Out]

d*x*(a+b*arccosh(c*x))-1/3*c^2*d*x^3*(a+b*arccosh(c*x))-7/9*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+1/9*b*c*d*x^2*(c
*x-1)^(1/2)*(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5894, 12, 471, 75} \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{3} c^2 d x^3 (a+b \text {arccosh}(c x))+d x (a+b \text {arccosh}(c x))+\frac {1}{9} b c d x^2 \sqrt {c x-1} \sqrt {c x+1}-\frac {7 b d \sqrt {c x-1} \sqrt {c x+1}}{9 c} \]

[In]

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

[Out]

(-7*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) + (b*c*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/9 + d*x*(a + b*ArcCosh[
c*x]) - (c^2*d*x^3*(a + b*ArcCosh[c*x]))/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 5894

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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = d x (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d x^3 (a+b \text {arccosh}(c x))-(b c) \int \frac {d x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d x (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d x^3 (a+b \text {arccosh}(c x))-(b c d) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{9} b c d x^2 \sqrt {-1+c x} \sqrt {1+c x}+d x (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d x^3 (a+b \text {arccosh}(c x))-\frac {1}{9} (7 b c d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+\frac {1}{9} b c d x^2 \sqrt {-1+c x} \sqrt {1+c x}+d x (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d x^3 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-7+c^2 x^2\right )+a \left (9 c x-3 c^3 x^3\right )-3 b c x \left (-3+c^2 x^2\right ) \text {arccosh}(c x)\right )}{9 c} \]

[In]

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

[Out]

(d*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-7 + c^2*x^2) + a*(9*c*x - 3*c^3*x^3) - 3*b*c*x*(-3 + c^2*x^2)*ArcCosh[c*x
]))/(9*c)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83

method	result	size
parts	\(-d a \left (\frac {1}{3} x^{3} c^{2}-x \right )-\frac {d b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(71\)
derivativedivides	\(\frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(73\)
default	\(\frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(73\)

[In]

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

[Out]

-d*a*(1/3*x^3*c^2-x)-d*b/c*(1/3*c^3*x^3*arccosh(c*x)-c*x*arccosh(c*x)-1/9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(c^2*x^2
-7))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {3 \, a c^{3} d x^{3} - 9 \, a c d x + 3 \, {\left (b c^{3} d x^{3} - 3 \, b c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d x^{2} - 7 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c} \]

[In]

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

[Out]

-1/9*(3*a*c^3*d*x^3 - 9*a*c*d*x + 3*(b*c^3*d*x^3 - 3*b*c*d*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^2*d*x^2 - 7*
b*d)*sqrt(c^2*x^2 - 1))/c

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{3} \, a c^{2} d x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d + a d x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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