3.1.87 \(\int \frac {\cosh ^{-1}(a+b x)}{x} \, dx\) [87]

Optimal. Leaf size=131 \[ -\frac {1}{2} \cosh ^{-1}(a+b x)^2+\cosh ^{-1}(a+b x) \log \left (1-\frac {e^{\cosh ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac {e^{\cosh ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )+\text {PolyLog}\left (2,\frac {e^{\cosh ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+\text {PolyLog}\left (2,\frac {e^{\cosh ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5996, 5962, 5681, 2221, 2317, 2438} \begin {gather*} \text {Li}_2\left (\frac {e^{\cosh ^{-1}(a+b x)}}{a-\sqrt {a^2-1}}\right )+\text {Li}_2\left (\frac {e^{\cosh ^{-1}(a+b x)}}{a+\sqrt {a^2-1}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac {e^{\cosh ^{-1}(a+b x)}}{a-\sqrt {a^2-1}}\right )+\cosh ^{-1}(a+b x) \log \left (1-\frac {e^{\cosh ^{-1}(a+b x)}}{\sqrt {a^2-1}+a}\right )-\frac {1}{2} \cosh ^{-1}(a+b x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a + b*x]/x,x]

[Out]

-1/2*ArcCosh[a + b*x]^2 + ArcCosh[a + b*x]*Log[1 - E^ArcCosh[a + b*x]/(a - Sqrt[-1 + a^2])] + ArcCosh[a + b*x]
*Log[1 - E^ArcCosh[a + b*x]/(a + Sqrt[-1 + a^2])] + PolyLog[2, E^ArcCosh[a + b*x]/(a - Sqrt[-1 + a^2])] + Poly
Log[2, E^ArcCosh[a + b*x]/(a + Sqrt[-1 + a^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 5681

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 5962

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]

Rule 5996

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 153, normalized size = 1.17 \begin {gather*} -\frac {1}{2} \cosh ^{-1}(a+b x)^2+\cosh ^{-1}(a+b x) \log \left (1+\frac {e^{\cosh ^{-1}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {-1+a^2}}{b}\right ) b}\right )+\cosh ^{-1}(a+b x) \log \left (1+\frac {e^{\cosh ^{-1}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {-1+a^2}}{b}\right ) b}\right )+\text {PolyLog}\left (2,-\frac {e^{\cosh ^{-1}(a+b x)}}{-a+\sqrt {-1+a^2}}\right )+\text {PolyLog}\left (2,\frac {e^{\cosh ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a + b*x]/x,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs. \(2(177)=354\).
time = 10.85, size = 431, normalized size = 3.29

method	result	size
derivativedivides	\(-\frac {\mathrm {arccosh}\left (b x +a \right )^{2}}{2}+\frac {a \,\mathrm {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {a \,\mathrm {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}+\frac {\left (a^{2}-1+a \sqrt {a^{2}-1}\right ) \mathrm {arccosh}\left (b x +a \right ) \left (\ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )+2 a \sqrt {a^{2}-1}\, \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )\right )}{a^{2}-1}+\dilog \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )+\dilog \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )\)	\(431\)
default	\(-\frac {\mathrm {arccosh}\left (b x +a \right )^{2}}{2}+\frac {a \,\mathrm {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}-\frac {a \,\mathrm {arccosh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}+\frac {\left (a^{2}-1+a \sqrt {a^{2}-1}\right ) \mathrm {arccosh}\left (b x +a \right ) \left (\ln \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )+2 a \sqrt {a^{2}-1}\, \ln \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )\right )}{a^{2}-1}+\dilog \left (\frac {\sqrt {a^{2}-1}+b x +\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{-a +\sqrt {a^{2}-1}}\right )+\dilog \left (\frac {\sqrt {a^{2}-1}-b x -\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{a +\sqrt {a^{2}-1}}\right )\)	\(431\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arccosh(b*x+a)^2+a*arccosh(b*x+a)/(a^2-1)^(1/2)*ln(((a^2-1)^(1/2)-b*x-(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/(a
+(a^2-1)^(1/2)))-a*arccosh(b*x+a)/(a^2-1)^(1/2)*ln(((a^2-1)^(1/2)+b*x+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/(-a+(a^
2-1)^(1/2)))+(a^2-1+a*(a^2-1)^(1/2))/(a^2-1)*arccosh(b*x+a)*(ln(((a^2-1)^(1/2)+b*x+(b*x+a-1)^(1/2)*(b*x+a+1)^(
1/2))/(-a+(a^2-1)^(1/2)))-2*ln(((a^2-1)^(1/2)-b*x-(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/(a+(a^2-1)^(1/2)))*a^2+ln((
(a^2-1)^(1/2)-b*x-(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/(a+(a^2-1)^(1/2)))+2*a*(a^2-1)^(1/2)*ln(((a^2-1)^(1/2)-b*x-
(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/(a+(a^2-1)^(1/2))))+dilog(((a^2-1)^(1/2)+b*x+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))
/(-a+(a^2-1)^(1/2)))+dilog(((a^2-1)^(1/2)-b*x-(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/(a+(a^2-1)^(1/2)))

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

[Out]

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

[Out]

integral(arccosh(b*x + a)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a + b*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(arccosh(b*x+a)/x,x, algorithm="giac")

[Out]

integrate(arccosh(b*x + a)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {acosh}\left (a+b\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a + b*x)/x,x)

[Out]

int(acosh(a + b*x)/x, x)

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