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

Optimal. Leaf size=58 \[ \frac{(a+b x) \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right )}{b}-\frac{2 c \tan ^{-1}\left (\sqrt{\frac{c \left (1-\frac{a}{c}\right )-b x}{a+b x+c}}\right )}{b} \]

[Out]

((a + b*x)*ArcSech[a/c + (b*x)/c])/b - (2*c*ArcTan[Sqrt[((1 - a/c)*c - b*x)/(a + c + b*x)]])/b

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Rubi [A]  time = 0.09298, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5893, 6313, 1961, 12, 203} \[ \frac{(a+b x) \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right )}{b}-\frac{2 c \tan ^{-1}\left (\sqrt{\frac{c \left (1-\frac{a}{c}\right )-b x}{a+b x+c}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c/(a + b*x)],x]

[Out]

((a + b*x)*ArcSech[a/c + (b*x)/c])/b - (2*c*ArcTan[Sqrt[((1 - a/c)*c - b*x)/(a + c + b*x)]])/b

Rule 5893

Int[ArcCosh[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcSech[a/c + (b*x^n)/c]^m, x] /
; FreeQ[{a, b, c, n, m}, x]

Rule 6313

Int[ArcSech[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[((c + d*x)*ArcSech[c + d*x])/d, x] + Int[Sqrt[(1 - c - d*x)/
(1 + c + d*x)]/(1 - c - d*x), x] /; FreeQ[{c, d}, x]

Rule 1961

Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n -
1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^(1/n + 1), x], x], x, ((e*(a + b*x
^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/
n] && IntegerQ[r]

Rule 12

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cosh ^{-1}\left (\frac{c}{a+b x}\right ) \, dx &=\int \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right ) \, dx\\ &=\frac{(a+b x) \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right )}{b}+\int \frac{\sqrt{\frac{1-\frac{a}{c}-\frac{b x}{c}}{1+\frac{a}{c}+\frac{b x}{c}}}}{1-\frac{a}{c}-\frac{b x}{c}} \, dx\\ &=\frac{(a+b x) \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right )}{b}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{c^2}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-\frac{a}{c}-\frac{b x}{c}}{1+\frac{a}{c}+\frac{b x}{c}}}\right )}{c}\\ &=\frac{(a+b x) \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right )}{b}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-\frac{a}{c}-\frac{b x}{c}}{1+\frac{a}{c}+\frac{b x}{c}}}\right )}{b}\\ &=\frac{(a+b x) \text{sech}^{-1}\left (\frac{a}{c}+\frac{b x}{c}\right )}{b}-\frac{2 c \tan ^{-1}\left (\sqrt{\frac{\left (1-\frac{a}{c}\right ) c-b x}{a+c+b x}}\right )}{b}\\ \end{align*}

Mathematica [B]  time = 0.42794, size = 175, normalized size = 3.02 \[ \frac{2 (a+b x-c)^{3/2} \left (a \sqrt{b} \sqrt{a+b x+c} \tan ^{-1}\left (\frac{\sqrt{a+b x-c}}{\sqrt{a+b x+c}}\right )-c \sqrt{b c} \sqrt{\frac{a+b x+c}{c}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{a+b x-c}}{\sqrt{2} \sqrt{b c}}\right )\right )}{b^{3/2} (a+b x)^2 \left (-\frac{a+b x-c}{a+b x}\right )^{3/2} \sqrt{\frac{a+b x+c}{a+b x}}}+x \cosh ^{-1}\left (\frac{c}{a+b x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[c/(a + b*x)],x]

[Out]

x*ArcCosh[c/(a + b*x)] + (2*(a - c + b*x)^(3/2)*(-(c*Sqrt[b*c]*Sqrt[(a + c + b*x)/c]*ArcSinh[(Sqrt[b]*Sqrt[a -
c + b*x])/(Sqrt[2]*Sqrt[b*c])]) + a*Sqrt[b]*Sqrt[a + c + b*x]*ArcTan[Sqrt[a - c + b*x]/Sqrt[a + c + b*x]]))/(
b^(3/2)*(a + b*x)^2*(-((a - c + b*x)/(a + b*x)))^(3/2)*Sqrt[(a + c + b*x)/(a + b*x)])

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Maple [A]  time = 0.043, size = 91, normalized size = 1.6 \begin{align*}{\rm arccosh} \left ({\frac{c}{bx+a}}\right )x+{\frac{a}{b}{\rm arccosh} \left ({\frac{c}{bx+a}}\right )}+{\frac{c}{b}\sqrt{{\frac{c}{bx+a}}-1}\sqrt{{\frac{c}{bx+a}}+1}\arctan \left ({\frac{1}{\sqrt{{\frac{{c}^{2}}{ \left ( bx+a \right ) ^{2}}}-1}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}}{ \left ( bx+a \right ) ^{2}}}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arccosh(c/(b*x+a))*x+1/b*arccosh(c/(b*x+a))*a+1/b*c*(c/(b*x+a)-1)^(1/2)*(c/(b*x+a)+1)^(1/2)/(1/(b*x+a)^2*c^2-1
)^(1/2)*arctan(1/(1/(b*x+a)^2*c^2-1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b x \log \left (\sqrt{b x + a + c} \sqrt{-b x - a + c} b x + \sqrt{b x + a + c} \sqrt{-b x - a + c} a +{\left (b x + a\right )} c\right ) - 2 \, b x \log \left (b x + a\right ) +{\left (a + c\right )} \log \left (b x + a + c\right ) - 2 \,{\left (b x + a\right )} \log \left (b x + a\right ) +{\left (a - c\right )} \log \left (-b x - a + c\right )}{2 \, b} + \int \frac{b^{2} c x^{2} + a b c x}{b^{2} c x^{2} + 2 \, a b c x + a^{2} c - c^{3} +{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + c\right ) + \frac{1}{2} \, \log \left (-b x - a + c\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*x*log(sqrt(b*x + a + c)*sqrt(-b*x - a + c)*b*x + sqrt(b*x + a + c)*sqrt(-b*x - a + c)*a + (b*x + a)*c
) - 2*b*x*log(b*x + a) + (a + c)*log(b*x + a + c) - 2*(b*x + a)*log(b*x + a) + (a - c)*log(-b*x - a + c))/b +
integrate((b^2*c*x^2 + a*b*c*x)/(b^2*c*x^2 + 2*a*b*c*x + a^2*c - c^3 + (b^2*x^2 + 2*a*b*x + a^2 - c^2)*e^(1/2*
log(b*x + a + c) + 1/2*log(-b*x - a + c))), x)

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Fricas [B]  time = 2.46134, size = 583, normalized size = 10.05 \begin{align*} \frac{2 \, b x \log \left (\frac{{\left (b x + a\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{b x + a}\right ) - 2 \, c \arctan \left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}\right ) + a \log \left (\frac{{\left (b x + a\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{x}\right ) - a \log \left (\frac{{\left (b x + a\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - c}{x}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*x*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - c^2)/(b^2*x^2 + 2*a*b*x + a^2)) + c)/(b*x + a)) - 2
*c*arctan((b^2*x^2 + 2*a*b*x + a^2)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - c^2)/(b^2*x^2 + 2*a*b*x + a^2))/(b^2*x^2
+ 2*a*b*x + a^2 - c^2)) + a*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - c^2)/(b^2*x^2 + 2*a*b*x + a^2)) +
c)/x) - a*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - c^2)/(b^2*x^2 + 2*a*b*x + a^2)) - c)/x))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(c/(a + b*x)), x)

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Giac [B]  time = 1.76533, size = 161, normalized size = 2.78 \begin{align*} -\frac{c \arcsin \left (-\frac{b x + a}{c}\right ) \mathrm{sgn}\left (b\right ) \mathrm{sgn}\left (c\right )}{{\left | b \right |}} + x \log \left (\sqrt{\frac{c}{b x + a} + 1} \sqrt{\frac{c}{b x + a} - 1} + \frac{c}{b x + a}\right ) + \frac{a \log \left (\frac{{\left | -2 \, b c - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + c^{2}}{\left | b \right |} \right |}}{{\left | -2 \, b^{2} x - 2 \, a b \right |}}\right )}{{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*arcsin(-(b*x + a)/c)*sgn(b)*sgn(c)/abs(b) + x*log(sqrt(c/(b*x + a) + 1)*sqrt(c/(b*x + a) - 1) + c/(b*x + a)
) + a*log(abs(-2*b*c - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + c^2)*abs(b))/abs(-2*b^2*x - 2*a*b))/abs(b)