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

Optimal. Leaf size=186 \[ -\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2} \]

[Out]

(-14*b^2*Sqrt[d - c^2*d*x^2])/(27*c^2) + (2*b^2*x^2*Sqrt[d - c^2*d*x^2])/27 + (2*b*x*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x]))/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*c^2*d)

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Rubi [A]  time = 0.370248, antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5798, 5718, 5680, 12, 460, 74} \[ -\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{(1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14*b^2*Sqrt[d - c^2*d*x^2])/(27*c^2) + (2*b^2*x^2*Sqrt[d - c^2*d*x^2])/27 + (2*b*x*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x]))/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^2)

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

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 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{\sqrt{d-c^2 d x^2} \int x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}-\frac{\left (2 b \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-3+c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-3+c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}-\frac{\left (14 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}+\frac{2 b x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{(1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.347932, size = 181, normalized size = 0.97 \[ \frac{\sqrt{d-c^2 d x^2} \left (9 a^2 \left (c^2 x^2-1\right )^2-6 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2-3\right )+6 b \cosh ^{-1}(c x) \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt{c x-1} \sqrt{c x+1} \left (3-c^2 x^2\right )\right )+2 b^2 \left (c^4 x^4-8 c^2 x^2+7\right )+9 b^2 \left (c^2 x^2-1\right )^2 \cosh ^{-1}(c x)^2\right )}{27 c^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(-6*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-3 + c^2*x^2) + 9*a^2*(-1 + c^2*x^2)^2 + 2*b^2*
(7 - 8*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - c^2*x^2) + 3*a*(-1 + c^2*x^2)^2)*ArcC
osh[c*x] + 9*b^2*(-1 + c^2*x^2)^2*ArcCosh[c*x]^2))/(27*c^2*(-1 + c^2*x^2))

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Maple [B]  time = 0.366, size = 726, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x)

[Out]

-1/3*a^2/c^2/d*(-c^2*d*x^2+d)^(3/2)+b^2*(1/216*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*
x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(9*arccosh(c*x)^2-6*arccosh(c*x)+2)/(c*x+1)/c^2/(c*x-1
)-1/8*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(arccosh(c*x)^2-2*arccosh(c*x)+2)/(c*
x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(arccosh(c*x)^2+2*arc
cosh(c*x)+2)/(c*x+1)/c^2/(c*x-1)+1/216*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^
4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(9*arccosh(c*x)^2+6*arccosh(c*x)+2)/(c*x+1)/c^2/(c*x-1))+2*a*
b*(1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x
-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/
2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/
2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)+1/72*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(
1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))/(c*x+1)/c^2/(c*x-1))

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Maxima [A]  time = 1.13214, size = 275, normalized size = 1.48 \begin{align*} \frac{2}{27} \, b^{2}{\left (\frac{\sqrt{c^{2} x^{2} - 1} \sqrt{-d} d x^{2} - \frac{7 \, \sqrt{c^{2} x^{2} - 1} \sqrt{-d} d}{c^{2}}}{d} - \frac{3 \,{\left (c^{2} \sqrt{-d} d x^{3} - 3 \, \sqrt{-d} d x\right )} \operatorname{arcosh}\left (c x\right )}{c d}\right )} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} b^{2} \operatorname{arcosh}\left (c x\right )^{2}}{3 \, c^{2} d} - \frac{2 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a b \operatorname{arcosh}\left (c x\right )}{3 \, c^{2} d} - \frac{2 \,{\left (c^{2} \sqrt{-d} d x^{3} - 3 \, \sqrt{-d} d x\right )} a b}{9 \, c d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a^{2}}{3 \, c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

2/27*b^2*((sqrt(c^2*x^2 - 1)*sqrt(-d)*d*x^2 - 7*sqrt(c^2*x^2 - 1)*sqrt(-d)*d/c^2)/d - 3*(c^2*sqrt(-d)*d*x^3 -
3*sqrt(-d)*d*x)*arccosh(c*x)/(c*d)) - 1/3*(-c^2*d*x^2 + d)^(3/2)*b^2*arccosh(c*x)^2/(c^2*d) - 2/3*(-c^2*d*x^2
+ d)^(3/2)*a*b*arccosh(c*x)/(c^2*d) - 2/9*(c^2*sqrt(-d)*d*x^3 - 3*sqrt(-d)*d*x)*a*b/(c*d) - 1/3*(-c^2*d*x^2 +
d)^(3/2)*a^2/(c^2*d)

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Fricas [A]  time = 2.27491, size = 591, normalized size = 3.18 \begin{align*} \frac{9 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (a b c^{3} x^{3} - 3 \, a b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 6 \,{\left ({\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 3 \,{\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \sqrt{-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - 2 \,{\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{27 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/27*(9*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(a*b*c^3*x
^3 - 3*a*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*((b^2*c^3*x^3 - 3*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqr
t(c^2*x^2 - 1) - 3*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + ((
9*a^2 + 2*b^2)*c^4*x^4 - 2*(9*a^2 + 8*b^2)*c^2*x^2 + 9*a^2 + 14*b^2)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acosh(c*x))**2*(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError