Integrand size = 10, antiderivative size = 58 \[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=\frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {2 c \arctan \left (\sqrt {\frac {\left (1-\frac {a}{c}\right ) c-b x}{a+c+b x}}\right )}{b} \]
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.00 \[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=x \text {arccosh}\left (\frac {c}{a+b x}\right )+\frac {2 \sqrt {-\frac {a-c+b x}{a+c+b x}} \sqrt {a+c+b x} \left (a \arctan \left (\frac {\sqrt {a-c+b x}}{\sqrt {a+c+b x}}\right )-c \text {arctanh}\left (\frac {\sqrt {a-c+b x}}{\sqrt {a+c+b x}}\right )\right )}{b \sqrt {a-c+b x}} \]
x*ArcCosh[c/(a + b*x)] + (2*Sqrt[-((a - c + b*x)/(a + c + b*x))]*Sqrt[a + c + b*x]*(a*ArcTan[Sqrt[a - c + b*x]/Sqrt[a + c + b*x]] - c*ArcTanh[Sqrt[a - c + b*x]/Sqrt[a + c + b*x]]))/(b*Sqrt[a - c + b*x])
Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6427, 6867, 2055, 27, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx\) |
\(\Big \downarrow \) 6427 |
\(\displaystyle \int \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )dx\) |
\(\Big \downarrow \) 6867 |
\(\displaystyle \int \frac {\sqrt {\frac {-\frac {a}{c}-\frac {b x}{c}+1}{\frac {a}{c}+\frac {b x}{c}+1}}}{-\frac {a}{c}-\frac {b x}{c}+1}dx+\frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}\) |
\(\Big \downarrow \) 2055 |
\(\displaystyle \frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {4 b \int \frac {c^2}{2 b^2 \left (\frac {\left (1-\frac {a}{c}\right ) c-b x}{a+c+b x}+1\right )}d\sqrt {\frac {\left (1-\frac {a}{c}\right ) c-b x}{a+c+b x}}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {2 c \int \frac {1}{\frac {\left (1-\frac {a}{c}\right ) c-b x}{a+c+b x}+1}d\sqrt {\frac {\left (1-\frac {a}{c}\right ) c-b x}{a+c+b x}}}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {2 c \arctan \left (\sqrt {\frac {c \left (1-\frac {a}{c}\right )-b x}{a+b x+c}}\right )}{b}\) |
((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
3.3.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/ (b*e - d*x^q)^(1/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1 /n))^r, 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] && Int egerQ[r]
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]
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] /; Free Q[{c, d}, x]
Time = 2.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.50
method | result | size |
derivativedivides | \(-\frac {c \left (-\frac {\left (b x +a \right ) \operatorname {arccosh}\left (\frac {c}{b x +a}\right )}{c}-\frac {\sqrt {\frac {c}{b x +a}-1}\, \sqrt {\frac {c}{b x +a}+1}\, \arctan \left (\frac {1}{\sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}-1}}\right )}{\sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}-1}}\right )}{b}\) | \(87\) |
default | \(-\frac {c \left (-\frac {\left (b x +a \right ) \operatorname {arccosh}\left (\frac {c}{b x +a}\right )}{c}-\frac {\sqrt {\frac {c}{b x +a}-1}\, \sqrt {\frac {c}{b x +a}+1}\, \arctan \left (\frac {1}{\sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}-1}}\right )}{\sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}-1}}\right )}{b}\) | \(87\) |
parts | \(x \,\operatorname {arccosh}\left (\frac {c}{b x +a}\right )+\frac {\sqrt {-\frac {b x +a -c}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +c}{b x +a}}\, \left (\ln \left (\frac {2 c \left (\operatorname {csgn}\left (c \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+c^{2}}+c \right ) b}{b x +a}\right ) \operatorname {csgn}\left (b \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-\left (b x +a +c \right ) \left (b x +a -c \right )}}\right ) \operatorname {csgn}\left (c \right ) c \right ) \operatorname {csgn}\left (c \right ) \operatorname {csgn}\left (b \right )}{b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+c^{2}}}\) | \(163\) |
-1/b*c*(-1/c*(b*x+a)*arccosh(c/(b*x+a))-(c/(b*x+a)-1)^(1/2)*(c/(b*x+a)+1)^ (1/2)/(c^2/(b*x+a)^2-1)^(1/2)*arctan(1/(c^2/(b*x+a)^2-1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (55) = 110\).
Time = 0.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 4.76 \[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=\frac {2 \, b x \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{b x + a}\right ) - 2 \, c \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}\right ) + a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{x}\right ) - a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - c}{x}\right )}{2 \, b} \]
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)*sq rt(-(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
\[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=\int \operatorname {acosh}{\left (\frac {c}{a + b x} \right )}\, dx \]
\[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (\frac {c}{b x + a}\right ) \,d x } \]
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 + in tegrate((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)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (55) = 110\).
Time = 2.54 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.05 \[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=\frac {c \arcsin \left (-\frac {b x + a}{c}\right ) \mathrm {sgn}\left (b\right ) \mathrm {sgn}\left (c\right )}{{\left | b \right |}} + x \log \left (\sqrt {\frac {c}{b x + a} + 1} \sqrt {\frac {c}{b x + a} - 1} + \frac {c}{b x + a}\right ) - \frac {a \log \left (\frac {{\left | -2 \, b c - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + c^{2}} {\left | b \right |} \right |}}{{\left | -2 \, b^{2} x - 2 \, a b \right |}}\right )}{{\left | b \right |}} \]
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)
Time = 4.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.91 \[ \int \text {arccosh}\left (\frac {c}{a+b x}\right ) \, dx=\frac {\mathrm {acosh}\left (\frac {c}{a+b\,x}\right )\,\left (a+b\,x\right )}{b}+\frac {c\,\mathrm {atan}\left (\frac {1}{\sqrt {\frac {c}{a+b\,x}-1}\,\sqrt {\frac {c}{a+b\,x}+1}}\right )}{b} \]