3.209 \(\int \frac{(a+b \cosh ^{-1}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}} \]

[Out]

(x*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(c*
d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(
c*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(c*d*Sqrt[d - c^2
*d*x^2])

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Rubi [A]  time = 0.351189, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5713, 5688, 5715, 3716, 2190, 2279, 2391} \[ -\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(3/2),x]

[Out]

(x*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(c*
d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(
c*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(c*d*Sqrt[d - c^2
*d*x^2])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5715

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.438871, size = 126, normalized size = 0.64 \[ \frac{\frac{\sqrt{c x-1} \sqrt{c x+1} \left (-2 b^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-2 b^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )+\left (a+b \cosh ^{-1}(c x)\right ) \left (a+b \cosh ^{-1}(c x)-2 b \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )\right )}{c}+x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(3/2),x]

[Out]

(x*(a + b*ArcCosh[c*x])^2 + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])*(a + b*ArcCosh[c*x] - 2*b*Log[
1 - E^ArcCosh[c*x]] - 2*b*Log[1 + E^ArcCosh[c*x]]) - 2*b^2*PolyLog[2, -E^ArcCosh[c*x]] - 2*b^2*PolyLog[2, E^Ar
cCosh[c*x]]))/c)/(d*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.216, size = 578, normalized size = 2.9 \begin{align*}{\frac{{a}^{2}x}{d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{ab\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right )x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{ab\sqrt{cx+1}\sqrt{cx-1}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}-1 \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a b c \sqrt{-\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{d} + b^{2} \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} + \frac{2 \, a b x \operatorname{arcosh}\left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a^{2} x}{\sqrt{-c^{2} d x^{2} + d} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

-a*b*c*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d*x^
2 + d)^(3/2), x) + 2*a*b*x*arccosh(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a^2*x/(sqrt(-c^2*d*x^2 + d)*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d
^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), 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}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/(-c^2*d*x^2 + d)^(3/2), x)