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

Optimal. Leaf size=117 \[ -\frac{1}{2} b d \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} b c d x \sqrt{c x-1} \sqrt{c x+1}-\frac{1}{4} b d \cosh ^{-1}(c x) \]

[Out]

(b*c*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/4 - (b*d*ArcCosh[c*x])/4 + (d*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/2 + (
d*(a + b*ArcCosh[c*x])^2)/(2*b) + d*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (b*d*PolyLog[2, -E^(-2
*ArcCosh[c*x])])/2

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Rubi [A]  time = 0.115941, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5727, 5660, 3718, 2190, 2279, 2391, 38, 52} \[ \frac{1}{2} b d \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} b c d x \sqrt{c x-1} \sqrt{c x+1}-\frac{1}{4} b d \cosh ^{-1}(c x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*c*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/4 - (b*d*ArcCosh[c*x])/4 + (d*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/2 - (
d*(a + b*ArcCosh[c*x])^2)/(2*b) + d*(a + b*ArcCosh[c*x])*Log[1 + E^(2*ArcCosh[c*x])] + (b*d*PolyLog[2, -E^(2*A
rcCosh[c*x])])/2

Rule 5727

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

Rule 5660

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

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && 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]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 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{\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+d \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx+\frac{1}{2} (b c d) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx\\ &=\frac{1}{4} b c d x \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+d \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )-\frac{1}{4} (b c d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} b c d x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{4} b d \cosh ^{-1}(c x)+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+(2 d) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{1}{4} b c d x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{4} b d \cosh ^{-1}(c x)+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-(b d) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac{1}{4} b c d x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{4} b d \cosh ^{-1}(c x)+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac{1}{2} (b d) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac{1}{4} b c d x \sqrt{-1+c x} \sqrt{1+c x}-\frac{1}{4} b d \cosh ^{-1}(c x)+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac{1}{2} b d \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.174166, size = 116, normalized size = 0.99 \[ -\frac{1}{4} d \left (2 b \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+2 a c^2 x^2-4 a \log (x)+2 b \cosh ^{-1}(c x) \left (c^2 x^2-2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-b c x \sqrt{c x-1} \sqrt{c x+1}-2 b \cosh ^{-1}(c x)^2-2 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(d*(2*a*c^2*x^2 - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*ArcCosh[c*x]^2 - 2*b*ArcTanh[Sqrt[(-1 + c*x)/(1 +
c*x)]] + 2*b*ArcCosh[c*x]*(c^2*x^2 - 2*Log[1 + E^(-2*ArcCosh[c*x])]) - 4*a*Log[x] + 2*b*PolyLog[2, -E^(-2*ArcC
osh[c*x])]))/4

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Maple [A]  time = 0.086, size = 131, normalized size = 1.1 \begin{align*} -{\frac{da{c}^{2}{x}^{2}}{2}}+da\ln \left ( cx \right ) -{\frac{db \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}-{\frac{db{\rm arccosh} \left (cx\right ){c}^{2}{x}^{2}}{2}}+{\frac{dbcx}{4}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{bd{\rm arccosh} \left (cx\right )}{4}}+db{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) +{\frac{db}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d*a*c^2*x^2+d*a*ln(c*x)-1/2*d*b*arccosh(c*x)^2-1/2*d*b*arccosh(c*x)*c^2*x^2+1/4*b*c*d*x*(c*x-1)^(1/2)*(c*
x+1)^(1/2)+1/4*b*d*arccosh(c*x)+d*b*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)+1/2*d*b*polylog(2,-
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*c^2*d*x^2 + a*d*log(x) - integrate(b*c^2*d*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) - b*d*log(c*x + sqr
t(c*x + 1)*sqrt(c*x - 1))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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