3.4 \(\int \frac {\cosh ^{-1}(c x)}{d+e x} \, dx\)

Optimal. Leaf size=178 \[ \frac {\text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\cosh ^{-1}(c x)^2}{2 e} \]

[Out]

-1/2*arccosh(c*x)^2/e+arccosh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e+arcco
sh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e+polylog(2,-e*(c*x+(c*x-1)^(1/2)*
(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e+polylog(2,-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^
(1/2)))/e

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5800, 5562, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}+\frac {\cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\cosh ^{-1}(c x)^2}{2 e} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*x]/(d + e*x),x]

[Out]

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

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 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 5800

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 176, normalized size = 0.99 \[ \frac {\text {Li}_2\left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )}{e}+\frac {\text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\cosh ^{-1}(c x)^2}{2 e} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[c*x]/(d + e*x),x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (c x\right )}{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d),x, algorithm="fricas")

[Out]

integral(arccosh(c*x)/(e*x + d), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (c x\right )}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d),x, algorithm="giac")

[Out]

integrate(arccosh(c*x)/(e*x + d), x)

________________________________________________________________________________________

maple [A]  time = 0.16, size = 295, normalized size = 1.66 \[ -\frac {\mathrm {arccosh}\left (c x \right )^{2}}{2 e}+\frac {\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {\dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {\dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(c*x)/(e*x+d),x)

[Out]

-1/2*arccosh(c*x)^2/e+1/e*arccosh(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e+c*d+(c^2*d^2-e^2)^(1/2))/(c*d+(
c^2*d^2-e^2)^(1/2)))+1/e*arccosh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e-c*d+(c^2*d^2-e^2)^(1/2))/(-c*d+
(c^2*d^2-e^2)^(1/2)))+1/e*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e-c*d+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-
e^2)^(1/2)))+1/e*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e+c*d+(c^2*d^2-e^2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2))
)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (c x\right )}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d),x, algorithm="maxima")

[Out]

integrate(arccosh(c*x)/(e*x + d), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (c\,x\right )}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)/(d + e*x),x)

[Out]

int(acosh(c*x)/(d + e*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (c x \right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*x)/(e*x+d),x)

[Out]

Integral(acosh(c*x)/(d + e*x), x)

________________________________________________________________________________________