3.2.37 \(\int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx\) [137]

Optimal. Leaf size=55 \[ \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} b \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5882, 3799, 2221, 2317, 2438} \begin {gather*} \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{2} b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/x,x]

[Out]

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 3799

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 5882

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.87 \begin {gather*} a \log (x)+\frac {1}{2} b \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )-\text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/x,x]

[Out]

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

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Maple [A]
time = 2.53, size = 75, normalized size = 1.36

method	result	size
derivativedivides	\(a \ln \left (c x \right )-\frac {b \mathrm {arccosh}\left (c x \right )^{2}}{2}+b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(75\)
default	\(a \ln \left (c x \right )-\frac {b \mathrm {arccosh}\left (c x \right )^{2}}{2}+b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x) + a*log(x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccosh(c*x) + a)/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c*x))/x, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))/x,x)

[Out]

int((a + b*acosh(c*x))/x, x)

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