3.1.5 \(\int \frac {\cosh ^{-1}(c x)}{(d+e x)^2} \, dx\) [5]

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

[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)

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Rubi [A]
time = 0.06, 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 = {5963, 95, 214} \begin {gather*} \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)} \end {gather*}

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 95

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 214

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

Rule 5963

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) \text {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*}

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Mathematica [A]
time = 0.07, size = 92, normalized size = 1.11 \begin {gather*} \frac {-\frac {\cosh ^{-1}(c x)}{d+e x}+\frac {c \left (\log (d+e x)-\log \left (e+c^2 d x-\sqrt {c^2 d^2-e^2} \sqrt {-1+c x} \sqrt {1+c x}\right )\right )}{\sqrt {c^2 d^2-e^2}}}{e} \end {gather*}

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

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Maple [A]
time = 5.67, size = 134, normalized size = 1.61

method	result	size
derivativedivides	\(\frac {-\frac {c^{2} \mathrm {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {c^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}}{c}\)	\(134\)
default	\(\frac {-\frac {c^{2} \mathrm {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {c^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}}{c}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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(c*d-%e>0)', see `assume?` for
more details

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (74) = 148\).
time = 0.39, size = 1023, normalized size = 12.33 \begin {gather*} \left [\frac {{\left (c d x \cosh \left (1\right ) + c d x \sinh \left (1\right ) + c d^{2}\right )} \sqrt {\frac {{\left (c^{2} d^{2} - 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} + 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}} \log \left (\frac {c^{3} d^{2} x + c d \cosh \left (1\right ) + c d \sinh \left (1\right ) + {\left (c^{2} d^{2} + c d \sqrt {\frac {{\left (c^{2} d^{2} - 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} + 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}} - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}\right )} \sqrt {c^{2} x^{2} - 1} + {\left (c^{2} d x + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \sqrt {\frac {{\left (c^{2} d^{2} - 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} + 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}}}{x \cosh \left (1\right ) + x \sinh \left (1\right ) + d}\right ) + {\left (c^{2} d^{2} x \cosh \left (1\right ) - x \cosh \left (1\right )^{3} - 3 \, x \cosh \left (1\right ) \sinh \left (1\right )^{2} - x \sinh \left (1\right )^{3} + {\left (c^{2} d^{2} x - 3 \, x \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (c^{2} d^{2} x \cosh \left (1\right ) + c^{2} d^{3} - x \cosh \left (1\right )^{3} - x \sinh \left (1\right )^{3} - d \cosh \left (1\right )^{2} - {\left (3 \, x \cosh \left (1\right ) + d\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d^{2} x - 3 \, x \cosh \left (1\right )^{2} - 2 \, d \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{3} x \cosh \left (1\right )^{2} + c^{2} d^{4} \cosh \left (1\right ) - d x \cosh \left (1\right )^{4} - d x \sinh \left (1\right )^{4} - d^{2} \cosh \left (1\right )^{3} - {\left (4 \, d x \cosh \left (1\right ) + d^{2}\right )} \sinh \left (1\right )^{3} + {\left (c^{2} d^{3} x - 6 \, d x \cosh \left (1\right )^{2} - 3 \, d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (2 \, c^{2} d^{3} x \cosh \left (1\right ) + c^{2} d^{4} - 4 \, d x \cosh \left (1\right )^{3} - 3 \, d^{2} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )}, -\frac {2 \, {\left (c d x \cosh \left (1\right ) + c d x \sinh \left (1\right ) + c d^{2}\right )} \sqrt {-\frac {{\left (c^{2} d^{2} - 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} + 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}} \arctan \left (-\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-\frac {{\left (c^{2} d^{2} - 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} + 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}} {\left (\cosh \left (1\right ) + \sinh \left (1\right )\right )} - {\left (c x \cosh \left (1\right ) + c x \sinh \left (1\right ) + c d\right )} \sqrt {-\frac {{\left (c^{2} d^{2} - 1\right )} \cosh \left (1\right ) - {\left (c^{2} d^{2} + 1\right )} \sinh \left (1\right )}{\cosh \left (1\right ) - \sinh \left (1\right )}}}{c^{2} d^{2} - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}\right ) - {\left (c^{2} d^{2} x \cosh \left (1\right ) - x \cosh \left (1\right )^{3} - 3 \, x \cosh \left (1\right ) \sinh \left (1\right )^{2} - x \sinh \left (1\right )^{3} + {\left (c^{2} d^{2} x - 3 \, x \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (c^{2} d^{2} x \cosh \left (1\right ) + c^{2} d^{3} - x \cosh \left (1\right )^{3} - x \sinh \left (1\right )^{3} - d \cosh \left (1\right )^{2} - {\left (3 \, x \cosh \left (1\right ) + d\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d^{2} x - 3 \, x \cosh \left (1\right )^{2} - 2 \, d \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{3} x \cosh \left (1\right )^{2} + c^{2} d^{4} \cosh \left (1\right ) - d x \cosh \left (1\right )^{4} - d x \sinh \left (1\right )^{4} - d^{2} \cosh \left (1\right )^{3} - {\left (4 \, d x \cosh \left (1\right ) + d^{2}\right )} \sinh \left (1\right )^{3} + {\left (c^{2} d^{3} x - 6 \, d x \cosh \left (1\right )^{2} - 3 \, d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (2 \, c^{2} d^{3} x \cosh \left (1\right ) + c^{2} d^{4} - 4 \, d x \cosh \left (1\right )^{3} - 3 \, d^{2} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[((c*d*x*cosh(1) + c*d*x*sinh(1) + c*d^2)*sqrt(((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh
(1)))*log((c^3*d^2*x + c*d*cosh(1) + c*d*sinh(1) + (c^2*d^2 + c*d*sqrt(((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*
sinh(1))/(cosh(1) - sinh(1))) - cosh(1)^2 - 2*cosh(1)*sinh(1) - sinh(1)^2)*sqrt(c^2*x^2 - 1) + (c^2*d*x + cosh
(1) + sinh(1))*sqrt(((c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1))))/(x*cosh(1) + x*sinh(
1) + d)) + (c^2*d^2*x*cosh(1) - x*cosh(1)^3 - 3*x*cosh(1)*sinh(1)^2 - x*sinh(1)^3 + (c^2*d^2*x - 3*x*cosh(1)^2
)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + (c^2*d^2*x*cosh(1) + c^2*d^3 - x*cosh(1)^3 - x*sinh(1)^3 - d*cosh(1)
^2 - (3*x*cosh(1) + d)*sinh(1)^2 + (c^2*d^2*x - 3*x*cosh(1)^2 - 2*d*cosh(1))*sinh(1))*log(-c*x + sqrt(c^2*x^2
- 1)))/(c^2*d^3*x*cosh(1)^2 + c^2*d^4*cosh(1) - d*x*cosh(1)^4 - d*x*sinh(1)^4 - d^2*cosh(1)^3 - (4*d*x*cosh(1)
+ d^2)*sinh(1)^3 + (c^2*d^3*x - 6*d*x*cosh(1)^2 - 3*d^2*cosh(1))*sinh(1)^2 + (2*c^2*d^3*x*cosh(1) + c^2*d^4 -
4*d*x*cosh(1)^3 - 3*d^2*cosh(1)^2)*sinh(1)), -(2*(c*d*x*cosh(1) + c*d*x*sinh(1) + c*d^2)*sqrt(-((c^2*d^2 - 1)
*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1)))*arctan(-(sqrt(c^2*x^2 - 1)*sqrt(-((c^2*d^2 - 1)*cosh(1)
- (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1)))*(cosh(1) + sinh(1)) - (c*x*cosh(1) + c*x*sinh(1) + c*d)*sqrt(-(
(c^2*d^2 - 1)*cosh(1) - (c^2*d^2 + 1)*sinh(1))/(cosh(1) - sinh(1))))/(c^2*d^2 - cosh(1)^2 - 2*cosh(1)*sinh(1)
- sinh(1)^2)) - (c^2*d^2*x*cosh(1) - x*cosh(1)^3 - 3*x*cosh(1)*sinh(1)^2 - x*sinh(1)^3 + (c^2*d^2*x - 3*x*cosh
(1)^2)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) - (c^2*d^2*x*cosh(1) + c^2*d^3 - x*cosh(1)^3 - x*sinh(1)^3 - d*co
sh(1)^2 - (3*x*cosh(1) + d)*sinh(1)^2 + (c^2*d^2*x - 3*x*cosh(1)^2 - 2*d*cosh(1))*sinh(1))*log(-c*x + sqrt(c^2
*x^2 - 1)))/(c^2*d^3*x*cosh(1)^2 + c^2*d^4*cosh(1) - d*x*cosh(1)^4 - d*x*sinh(1)^4 - d^2*cosh(1)^3 - (4*d*x*co
sh(1) + d^2)*sinh(1)^3 + (c^2*d^3*x - 6*d*x*cosh(1)^2 - 3*d^2*cosh(1))*sinh(1)^2 + (2*c^2*d^3*x*cosh(1) + c^2*
d^4 - 4*d*x*cosh(1)^3 - 3*d^2*cosh(1)^2)*sinh(1))]

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

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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (71) = 142\).
time = 0.58, size = 240, normalized size = 2.89 \begin {gather*} -\frac {\log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{{\left (e x + d\right )} e} + \frac {\frac {c e^{4} \log \left ({\left | c^{2} d e - \sqrt {c^{2} d^{2} - e^{2}} {\left | c \right |} {\left | e \right |} \right |}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c^{2} d^{2} - e^{2}} {\left | e \right |}} - \frac {c e^{4} \log \left ({\left | c^{2} d e - \sqrt {c^{2} d^{2} - e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{e x + d} + \frac {c^{2} d^{2}}{{\left (e x + d\right )}^{2}} - \frac {e^{2}}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} - e^{4}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |} \right |}\right )}{\sqrt {c^{2} d^{2} - e^{2}} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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