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

Optimal. Leaf size=74 \[ -\frac{b \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^2 d}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.116287, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5715, 3716, 2190, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 c^2 d}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0858123, size = 85, normalized size = 1.15 \[ \frac{-2 b^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-2 b^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )+\left (a+b \cosh ^{-1}(c x)\right ) \left (a+b \cosh ^{-1}(c x)-2 b \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )}{2 b c^2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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^ArcCosh[c*x]])/(2*b*c^2*d)

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 179, normalized size = 2.4 \begin{align*} -{\frac{a\ln \left ( cx-1 \right ) }{2\,{c}^{2}d}}-{\frac{a\ln \left ( cx+1 \right ) }{2\,{c}^{2}d}}+{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2\,{c}^{2}d}}-{\frac{b{\rm arccosh} \left (cx\right )}{{c}^{2}d}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{{c}^{2}d}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b{\rm arccosh} \left (cx\right )}{{c}^{2}d}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{b}{{c}^{2}d}{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, b{\left (\frac{4 \,{\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - \log \left (c x + 1\right )^{2} - 2 \, \log \left (c x + 1\right ) \log \left (c x - 1\right ) - \log \left (c x - 1\right )^{2}}{c^{2} d} + 8 \, \int \frac{\log \left (c x + 1\right ) + \log \left (c x - 1\right )}{2 \,{\left (c^{4} d x^{3} - c^{2} d x +{\left (c^{3} d x^{2} - c d\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x}\right )} - \frac{a \log \left (c^{2} d x^{2} - d\right )}{2 \, c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*((4*(log(c*x + 1) + log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) - log(c*x + 1)^2 - 2*log(c*x +
1)*log(c*x - 1) - log(c*x - 1)^2)/(c^2*d) + 8*integrate(1/2*(log(c*x + 1) + log(c*x - 1))/(c^4*d*x^3 - c^2*d*
x + (c^3*d*x^2 - c*d)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) - 1/2*a*log(c^2*d*x^2 - d)/(c^2*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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