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

Optimal. Leaf size=226 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d-c^2 d x^2}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{b^2 x (1-c x) (c x+1)}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}} \]

[Out]

-(b^2*x*(1 - c*x)*(1 + c*x))/(4*c^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(4*
c^3*Sqrt[d - c^2*d*x^2]) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c*Sqrt[d - c^2*d*x^2])
- (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*c^2*d) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x
])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2])

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Rubi [A]  time = 0.626361, antiderivative size = 234, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {5798, 5759, 5676, 5662, 90, 52} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d-c^2 d x^2}}-\frac{x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{b^2 x (1-c x) (c x+1)}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*x*(1 - c*x)*(1 + c*x))/(4*c^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(4*
c^3*Sqrt[d - c^2*d*x^2]) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c*Sqrt[d - c^2*d*x^2])
- (x*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x])^2)/(2*c^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*
(a + b*ArcCosh[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^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 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5676

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

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{c \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x (1-c x) (1+c x)}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x (1-c x) (1+c x)}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c \sqrt{d-c^2 d x^2}}-\frac{x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.862718, size = 228, normalized size = 1.01 \[ \frac{-\frac{12 a^2 c x \sqrt{d-c^2 d x^2}}{d}-\frac{12 a^2 \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{\sqrt{d}}+\frac{6 a b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+\sinh \left (2 \cosh ^{-1}(c x)\right )\right )-\cosh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (4 \cosh ^{-1}(c x)^3-6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+\left (6 \cosh ^{-1}(c x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt{d-c^2 d x^2}}}{24 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-12*a^2*c*x*Sqrt[d - c^2*d*x^2])/d - (12*a^2*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqr
t[d] + (b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(4*ArcCosh[c*x]^3 - 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + (3
+ 6*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2] + (6*a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(
-Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] + Sinh[2*ArcCosh[c*x]])))/Sqrt[d - c^2*d*x^2])/(24*c^3)

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Maple [B]  time = 0.309, size = 624, normalized size = 2.8 \begin{align*} -{\frac{{a}^{2}x}{2\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{{a}^{2}}{2\,{c}^{2}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{{b}^{2}{x}^{3}}{4\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}x}{4\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{3}}{6\,{c}^{3}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{{b}^{2}{\rm arccosh} \left (cx\right ){x}^{2}}{2\,cd \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{x}^{3}}{2\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}x}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2}{\rm arccosh} \left (cx\right )}{4\,{c}^{3}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{ab \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2\,{c}^{3}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{ab{\rm arccosh} \left (cx\right ){x}^{3}}{d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{ab{x}^{2}}{2\,cd \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{ab{\rm arccosh} \left (cx\right )x}{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{ab}{4\,{c}^{3}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a^2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1
/4*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*x^3+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)/d/c^2/(c^2*x^2-1)*x-1/6*b^2*(-d
*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)^3+1/2*b^2*(-d*(c^2*x^2-1))^(1/2
)/d/c/(c^2*x^2-1)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*ar
ccosh(c*x)^2*x^3+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)/d/c^2/(c^2*x^2-1)*arccosh(c*x)^2*x-1/4*b^2*(-d*(c^2*x^2-1))^(1
/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)-1/2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c
*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)^2-a*b*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)*x^3+1/2*a*b
*(-d*(c^2*x^2-1))^(1/2)/d/c/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+a*b*(-d*(c^2*x^2-1))^(1/2)/d/c^2/(c^2*
x^2-1)*arccosh(c*x)*x-1/4*a*b*(-d*(c^2*x^2-1))^(1/2)/d/c^3/(c^2*x^2-1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b^{2} x^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arcosh}\left (c x\right ) + a^{2} x^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b^2*x^2*arccosh(c*x)^2 + 2*a*b*x^2*arccosh(c*x) + a^2*x^2)*sqrt(-c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2*x^2/sqrt(-c^2*d*x^2 + d), x)