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

Optimal. Leaf size=135 \[ \frac{1}{2} b c^2 d \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-\frac{d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac{c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{2} b c^2 d \cosh ^{-1}(c x)+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.128991, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5729, 97, 12, 52, 5660, 3718, 2190, 2279, 2391} \[ -\frac{1}{2} b c^2 d \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )-\frac{d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{2} b c^2 d \cosh ^{-1}(c x)+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{2 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 5729

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

Rule 97

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.154572, size = 106, normalized size = 0.79 \[ -\frac{d \left (-b c^2 x^2 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+2 a c^2 x^2 \log (x)+a+b c^2 x^2 \cosh ^{-1}(c x)^2+b \cosh ^{-1}(c x) \left (2 c^2 x^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+1\right )-b c x \sqrt{c x-1} \sqrt{c x+1}\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.157, size = 140, normalized size = 1. \begin{align*} -{c}^{2}da\ln \left ( cx \right ) -{\frac{da}{2\,{x}^{2}}}+{\frac{{c}^{2}db \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}+{\frac{bcd}{2\,x}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{c}^{2}db}{2}}-{\frac{bd{\rm arccosh} \left (cx\right )}{2\,{x}^{2}}}-{c}^{2}db{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{\frac{{c}^{2}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^3,x)

[Out]

-c^2*d*a*ln(c*x)-1/2*d*a/x^2+1/2*c^2*d*b*arccosh(c*x)^2+1/2*b*c*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x-1/2*c^2*d*b-1/
2*d*b*arccosh(c*x)/x^2-c^2*d*b*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)-1/2*c^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*} -b c^{2} d \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{x}\,{d x} - a c^{2} d \log \left (x\right ) + \frac{1}{2} \, b d{\left (\frac{\sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\operatorname{arcosh}\left (c x\right )}{x^{2}}\right )} - \frac{a d}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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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^{3}}, 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^3,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^3, x)

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

[Out]

-d*(Integral(-a/x**3, x) + Integral(a*c**2/x, x) + Integral(-b*acosh(c*x)/x**3, x) + Integral(b*c**2*acosh(c*x
)/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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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