3.5 \(\int \frac {\cosh ^{-1}(c x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=83 \[ \frac {2 c \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{e \sqrt {c d-e} \sqrt {c d+e}}-\frac {\cosh ^{-1}(c x)}{e (d+e x)} \]

[Out]

-arccosh(c*x)/e/(e*x+d)+2*c*arctanh((c*d+e)^(1/2)*(c*x+1)^(1/2)/(c*d-e)^(1/2)/(c*x-1)^(1/2))/e/(c*d-e)^(1/2)/(
c*d+e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5802, 93, 208} \[ \frac {2 c \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{e \sqrt {c d-e} \sqrt {c d+e}}-\frac {\cosh ^{-1}(c x)}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*x]/(d + e*x)^2,x]

[Out]

-(ArcCosh[c*x]/(e*(d + e*x))) + (2*c*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(S
qrt[c*d - e]*e*Sqrt[c*d + e])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rule 5802

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

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac {\cosh ^{-1}(c x)}{e (d+e x)}+\frac {c \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{e}\\ &=-\frac {\cosh ^{-1}(c x)}{e (d+e x)}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{c d-e-(c d+e) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{e}\\ &=-\frac {\cosh ^{-1}(c x)}{e (d+e x)}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{\sqrt {c d-e} e \sqrt {c d+e}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 92, normalized size = 1.11 \[ \frac {\frac {c \left (\log (d+e x)-\log \left (-\sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d^2-e^2}+c^2 d x+e\right )\right )}{\sqrt {c^2 d^2-e^2}}-\frac {\cosh ^{-1}(c x)}{d+e x}}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[c*x]/(d + e*x)^2,x]

[Out]

(-(ArcCosh[c*x]/(d + e*x)) + (c*(Log[d + e*x] - Log[e + c^2*d*x - Sqrt[c^2*d^2 - e^2]*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]]))/Sqrt[c^2*d^2 - e^2])/e

________________________________________________________________________________________

fricas [B]  time = 0.56, size = 454, normalized size = 5.47 \[ \left [\frac {{\left (c^{2} d^{2} e - e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} d^{2} - e^{2}} {\left (c d e x + c d^{2}\right )} \log \left (\frac {c^{3} d^{2} x + c d e + \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + {\left (c^{2} d^{3} - d e^{2} + {\left (c^{2} d^{2} e - e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, \frac {{\left (c^{2} d^{2} e - e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c d e x + c d^{2}\right )} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + {\left (c^{2} d^{3} - d e^{2} + {\left (c^{2} d^{2} e - e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[((c^2*d^2*e - e^3)*x*log(c*x + sqrt(c^2*x^2 - 1)) + sqrt(c^2*d^2 - e^2)*(c*d*e*x + c*d^2)*log((c^3*d^2*x + c*
d*e + sqrt(c^2*d^2 - e^2)*(c^2*d*x + e) + (c^2*d^2 + sqrt(c^2*d^2 - e^2)*c*d - e^2)*sqrt(c^2*x^2 - 1))/(e*x +
d)) + (c^2*d^3 - d*e^2 + (c^2*d^2*e - e^3)*x)*log(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^4*e - d^2*e^3 + (c^2*d^3*e
^2 - d*e^4)*x), ((c^2*d^2*e - e^3)*x*log(c*x + sqrt(c^2*x^2 - 1)) - 2*sqrt(-c^2*d^2 + e^2)*(c*d*e*x + c*d^2)*a
rctan(-(sqrt(-c^2*d^2 + e^2)*sqrt(c^2*x^2 - 1)*e - sqrt(-c^2*d^2 + e^2)*(c*e*x + c*d))/(c^2*d^2 - e^2)) + (c^2
*d^3 - d*e^2 + (c^2*d^2*e - e^3)*x)*log(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^4*e - d^2*e^3 + (c^2*d^3*e^2 - d*e^4
)*x)]

________________________________________________________________________________________

giac [B]  time = 1.15, size = 215, normalized size = 2.59 \[ {\left (\frac {c e^{3} \log \left ({\left | c^{2} d - \sqrt {c^{2} d^{2} - e^{2}} {\left | c \right |} \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c^{2} d^{2} - e^{2}}} - \frac {c e^{3} \log \left ({\left | c^{2} d - \sqrt {c^{2} d^{2} - e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{x e + d} + \frac {c^{2} d^{2}}{{\left (x e + d\right )}^{2}} - \frac {e^{2}}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} - e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{\sqrt {c^{2} d^{2} - e^{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-4\right )} - \frac {e^{\left (-1\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{x e + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*e^3*log(abs(c^2*d - sqrt(c^2*d^2 - e^2)*abs(c)))*sgn(1/(x*e + d))/sqrt(c^2*d^2 - e^2) - c*e^3*log(abs(c^2*d
- sqrt(c^2*d^2 - e^2)*(sqrt(c^2 - 2*c^2*d/(x*e + d) + c^2*d^2/(x*e + d)^2 - e^2/(x*e + d)^2) + sqrt(c^2*d^2*e
^2 - e^4)*e^(-1)/(x*e + d))))/(sqrt(c^2*d^2 - e^2)*sgn(1/(x*e + d))))*e^(-4) - e^(-1)*log(c*x + sqrt(c^2*x^2 -
1))/(x*e + d)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 126, normalized size = 1.52 \[ -\frac {c \,\mathrm {arccosh}\left (c x \right )}{\left (c x e +c d \right ) e}-\frac {c \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(c*x)/(e*x+d)^2,x)

[Out]

-c/(c*e*x+c*d)/e*arccosh(c*x)-c/e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2
)/e^2)^(1/2)*e+e)/(c*e*x+c*d))/((c^2*d^2-e^2)/e^2)^(1/2)/(c^2*x^2-1)^(1/2)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c*d>0)', see `assume?` for m
ore details)Is e-c*d positive, negative or zero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)/(d + e*x)^2,x)

[Out]

int(acosh(c*x)/(d + e*x)^2, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*x)/(e*x+d)**2,x)

[Out]

Integral(acosh(c*x)/(d + e*x)**2, x)

________________________________________________________________________________________