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

Optimal. Leaf size=59 \[ \frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c d}-\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c d}+\frac{2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d} \]

[Out]

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

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Rubi [A]  time = 0.0653781, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5694, 4182, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c d}-\frac{b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c d}+\frac{2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5694

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c d}\\ &=\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c d}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c d}-\frac{b \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c d}\\ &=\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c d}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c d}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c d}\\ &=\frac{2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c d}+\frac{b \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c d}-\frac{b \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c d}\\ \end{align*}

Mathematica [A]  time = 0.0668811, size = 64, normalized size = 1.08 \[ \frac{b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )+\left (\log \left (1-e^{\cosh ^{-1}(c x)}\right )-\log \left (e^{\cosh ^{-1}(c x)}+1\right )\right ) \left (-\left (a+b \cosh ^{-1}(c x)\right )\right )}{c d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.283, size = 338, normalized size = 5.7 \begin{align*}{\frac{a{\it Artanh} \left ( cx \right ) }{cd}}+{\frac{b{\it Artanh} \left ( cx \right ){\rm arccosh} \left (cx\right )}{cd}}+{\frac{2\,ib{\it Artanh} \left ( cx \right ) }{cd \left ( cx-1 \right ) \left ( cx+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{cx}{2}}}\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{cx}{2}}}\ln \left ( 1+{i \left ( cx+1 \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{2\,ib{\it Artanh} \left ( cx \right ) }{cd \left ( cx-1 \right ) \left ( cx+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{cx}{2}}}\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{cx}{2}}}\ln \left ( 1-{i \left ( cx+1 \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{2\,ib}{cd \left ( cx-1 \right ) \left ( cx+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{cx}{2}}}\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{cx}{2}}}{\it dilog} \left ( 1+{i \left ( cx+1 \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{2\,ib}{cd \left ( cx-1 \right ) \left ( cx+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{cx}{2}}}\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{cx}{2}}}{\it dilog} \left ( 1-{i \left ( cx+1 \right ){\frac{1}{\sqrt{-{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))/(-c^2*d*x^2+d),x)

[Out]

1/c*a/d*arctanh(c*x)+1/c*b/d*arctanh(c*x)*arccosh(c*x)+2*I/c*b/d*(1/2+1/2*c*x)^(1/2)*(-c^2*x^2+1)^(1/2)*(-1/2+
1/2*c*x)^(1/2)/(c*x-1)/(c*x+1)*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-2*I/c*b/d*(1/2+1/2*c*x)^(1/2)*(
-c^2*x^2+1)^(1/2)*(-1/2+1/2*c*x)^(1/2)/(c*x-1)/(c*x+1)*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+2*I/c*b
/d*(1/2+1/2*c*x)^(1/2)*(-c^2*x^2+1)^(1/2)*(-1/2+1/2*c*x)^(1/2)/(c*x-1)/(c*x+1)*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^
(1/2))-2*I/c*b/d*(1/2+1/2*c*x)^(1/2)*(-c^2*x^2+1)^(1/2)*(-1/2+1/2*c*x)^(1/2)/(c*x-1)/(c*x+1)*dilog(1-I*(c*x+1)
/(-c^2*x^2+1)^(1/2))

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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 )}{c d} + 8 \, \int \frac{{\left (3 \, c x - 1\right )} \log \left (c x - 1\right )}{4 \,{\left (c^{2} d x^{2} - d\right )}}\,{d x} + 8 \, \int \frac{\log \left (c x + 1\right ) - \log \left (c x - 1\right )}{2 \,{\left (c^{3} d x^{3} - c d x +{\left (c^{2} d x^{2} - d\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}}\,{d x}\right )} + \frac{1}{2} \, a{\left (\frac{\log \left (c x + 1\right )}{c d} - \frac{\log \left (c x - 1\right )}{c d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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