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

Optimal. Leaf size=94 \[ d x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (9 c^2 d+2 e\right )}{9 c^3}-\frac{b e x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]

[Out]

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

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Rubi [A]  time = 0.0799358, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5705, 460, 74} \[ d x \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (9 c^2 d+2 e\right )}{9 c^3}-\frac{b e x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(9*c^2*d + 2*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*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 5705

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])

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+e 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} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-(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 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-\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 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0855492, size = 76, normalized size = 0.81 \[ \frac{1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (9 d+e x^2\right )+2 e\right )}{c^3}+3 b x \cosh ^{-1}(c x) \left (3 d+e x^2\right )\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.008, size = 90, normalized size = 1. \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{2}} \left ({\frac{{c}^{3}{x}^{3}e}{3}}+{c}^{3}dx \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{3}{x}^{3}e}{3}}+{\rm arccosh} \left (cx\right ){c}^{3}dx-{\frac{{x}^{2}{c}^{2}e+9\,{c}^{2}d+2\,e}{9}\sqrt{cx-1}\sqrt{cx+1}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08782, size = 123, normalized size = 1.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.3047, size = 208, normalized size = 2.21 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.911373, size = 116, normalized size = 1.23 \begin{align*} \begin{cases} a d x + \frac{a e x^{3}}{3} + b d x \operatorname{acosh}{\left (c x \right )} + \frac{b e x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{b d \sqrt{c^{2} x^{2} - 1}}{c} - \frac{b e x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{2 b e \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (d x + \frac{e x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23011, size = 146, normalized size = 1.55 \begin{align*}{\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 + \frac{1}{9} \,{\left (3 \, a x^{3} +{\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\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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