3.5 \(\int (d-c^2 d x^2) (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=86 \[ -\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )+d x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} b c d x^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]

[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

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Rubi [A]  time = 0.074217, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5680, 12, 460, 74} \[ -\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )+d x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} b c d x^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]

Antiderivative was successfully verified.

[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 5680

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]

Rule 12

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

Rule 460

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 74

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]

Rubi steps

\begin{align*} \int \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d x \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )-(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 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )-(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 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\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 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} c^2 d x^3 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0845899, size = 71, normalized size = 0.83 \[ \frac{d \left (a \left (9 c x-3 c^3 x^3\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2-7\right )-3 b c x \left (c^2 x^2-3\right ) \cosh ^{-1}(c x)\right )}{9 c} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.011, size = 73, normalized size = 0.9 \begin{align*}{\frac{1}{c} \left ( -da \left ({\frac{{c}^{3}{x}^{3}}{3}}-cx \right ) -db \left ({\frac{{c}^{3}{x}^{3}{\rm arccosh} \left (cx\right )}{3}}-cx{\rm arccosh} \left (cx\right )-{\frac{{c}^{2}{x}^{2}-7}{9}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-d*a*(1/3*c^3*x^3-c*x)-d*b*(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)))

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Maxima [A]  time = 1.18129, size = 131, normalized size = 1.52 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.82291, size = 185, normalized size = 2.15 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.755015, size = 97, normalized size = 1.13 \begin{align*} \begin{cases} - \frac{a c^{2} d x^{3}}{3} + a d x - \frac{b c^{2} d x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{b c d x^{2} \sqrt{c^{2} x^{2} - 1}}{9} + b d x \operatorname{acosh}{\left (c x \right )} - \frac{7 b d \sqrt{c^{2} x^{2} - 1}}{9 c} & \text{for}\: c \neq 0 \\d x \left (a + \frac{i \pi b}{2}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**2*d*x**3/3 + a*d*x - b*c**2*d*x**3*acosh(c*x)/3 + b*c*d*x**2*sqrt(c**2*x**2 - 1)/9 + b*d*x*ac
osh(c*x) - 7*b*d*sqrt(c**2*x**2 - 1)/(9*c), Ne(c, 0)), (d*x*(a + I*pi*b/2), True))

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Giac [A]  time = 1.37603, size = 151, normalized size = 1.76 \begin{align*} -\frac{1}{3} \, a c^{2} d x^{3} - \frac{1}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b c^{2} d +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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