3.1.6 \(\int \frac {\cosh ^{-1}(c x)}{(d+e x)^3} \, dx\) [6]

Optimal. Leaf size=132 \[ -\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {c^3 d \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}} \]

[Out]

-1/2*arccosh(c*x)/e/(e*x+d)^2+c^3*d*arctanh((c*d+e)^(1/2)*(c*x+1)^(1/2)/(c*d-e)^(1/2)/(c*x-1)^(1/2))/(c*d-e)^(
3/2)/e/(c*d+e)^(3/2)-1/2*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)

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Rubi [A]
time = 0.10, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5963, 98, 95, 214} \begin {gather*} \frac {c^3 d \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{e (c d-e)^{3/2} (c d+e)^{3/2}}-\frac {c \sqrt {c x-1} \sqrt {c x+1}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*x]/(d + e*x)^3,x]

[Out]

-1/2*(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d^2 - e^2)*(d + e*x)) - ArcCosh[c*x]/(2*e*(d + e*x)^2) + (c^3*d*Ar
cTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/((c*d - e)^(3/2)*e*(c*d + e)^(3/2))

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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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)^3} \, dx &=-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {c \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{2 e}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (c^3 d\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{2 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (c^3 d\right ) \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 \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {c^3 d \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 190, normalized size = 1.44 \begin {gather*} \frac {-\left (c^2 d^2-e^2\right )^{3/2} \cosh ^{-1}(c x)+c (d+e x) \left (-e \sqrt {c^2 d^2-e^2} \sqrt {-1+c x} \sqrt {1+c x}+c^2 d (d+e x) \log (d+e x)-c^2 d (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 )}{2 (c d-e) e (c d+e) \sqrt {c^2 d^2-e^2} (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[c*x]/(d + e*x)^3,x]

[Out]

(-((c^2*d^2 - e^2)^(3/2)*ArcCosh[c*x]) + c*(d + e*x)*(-(e*Sqrt[c^2*d^2 - e^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) +
c^2*d*(d + e*x)*Log[d + e*x] - c^2*d*(d + e*x)*Log[e + c^2*d*x - Sqrt[c^2*d^2 - e^2]*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]]))/(2*(c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2]*(d + e*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(112)=224\).
time = 5.73, size = 266, normalized size = 2.02

method	result	size
derivativedivides	\(\frac {-\frac {c^{3} \mathrm {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {c^{3} \left (-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d -e \right ) \left (c d +e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(266\)
default	\(\frac {-\frac {c^{3} \mathrm {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {c^{3} \left (-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d -e \right ) \left (c d +e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(c*x)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/2*c^3/(c*e*x+c*d)^2/e*arccosh(c*x)+1/2*c^3/e^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-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*e*x-e^2*(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d-e
)/(c*d+e)/(c*e*x+c*d)/((c^2*d^2-e^2)/e^2)^(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)^3,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. 1882 vs. \(2 (115) = 230\).
time = 0.45, size = 3784, normalized size = 28.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[-1/2*(2*c^4*d^5*x*cosh(1) + c^4*d^6 - c^2*d^2*x^2*cosh(1)^4 - c^2*d^2*x^2*sinh(1)^4 - 2*c^2*d^3*x*cosh(1)^3 -
2*(2*c^2*d^2*x^2*cosh(1) + c^2*d^3*x)*sinh(1)^3 + (c^4*d^4*x^2 - c^2*d^4)*cosh(1)^2 + (c^4*d^4*x^2 - 6*c^2*d^
2*x^2*cosh(1)^2 - 6*c^2*d^3*x*cosh(1) - c^2*d^4)*sinh(1)^2 + (c^3*d^3*x^2*cosh(1)^2 + c^3*d^3*x^2*sinh(1)^2 +
2*c^3*d^4*x*cosh(1) + c^3*d^5 + 2*(c^3*d^3*x^2*cosh(1) + c^3*d^4*x)*sinh(1))*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^4*d^4*x^2*cosh(1)^2 + 2*c^4*d^5*x*cosh(1) - 2*c^2*d^2*x^2*cos
h(1)^4 - 4*c^2*d^3*x*cosh(1)^3 + x^2*cosh(1)^6 + x^2*sinh(1)^6 + 2*d*x*cosh(1)^5 + 2*(3*x^2*cosh(1) + d*x)*sin
h(1)^5 - (2*c^2*d^2*x^2 - 15*x^2*cosh(1)^2 - 10*d*x*cosh(1))*sinh(1)^4 - 4*(2*c^2*d^2*x^2*cosh(1) + c^2*d^3*x
- 5*x^2*cosh(1)^3 - 5*d*x*cosh(1)^2)*sinh(1)^3 + (c^4*d^4*x^2 - 12*c^2*d^2*x^2*cosh(1)^2 - 12*c^2*d^3*x*cosh(1
) + 15*x^2*cosh(1)^4 + 20*d*x*cosh(1)^3)*sinh(1)^2 + 2*(c^4*d^4*x^2*cosh(1) + c^4*d^5*x - 4*c^2*d^2*x^2*cosh(1
)^3 - 6*c^2*d^3*x*cosh(1)^2 + 3*x^2*cosh(1)^5 + 5*d*x*cosh(1)^4)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) - (2*c^
4*d^5*x*cosh(1) + c^4*d^6 - 4*c^2*d^3*x*cosh(1)^3 + x^2*cosh(1)^6 + x^2*sinh(1)^6 + 2*d*x*cosh(1)^5 + 2*(3*x^2
*cosh(1) + d*x)*sinh(1)^5 - (2*c^2*d^2*x^2 - d^2)*cosh(1)^4 - (2*c^2*d^2*x^2 - 15*x^2*cosh(1)^2 - 10*d*x*cosh(
1) - d^2)*sinh(1)^4 - 4*(c^2*d^3*x - 5*x^2*cosh(1)^3 - 5*d*x*cosh(1)^2 + (2*c^2*d^2*x^2 - d^2)*cosh(1))*sinh(1
)^3 + (c^4*d^4*x^2 - 2*c^2*d^4)*cosh(1)^2 + (c^4*d^4*x^2 - 12*c^2*d^3*x*cosh(1) - 2*c^2*d^4 + 15*x^2*cosh(1)^4
+ 20*d*x*cosh(1)^3 - 6*(2*c^2*d^2*x^2 - d^2)*cosh(1)^2)*sinh(1)^2 + 2*(c^4*d^5*x - 6*c^2*d^3*x*cosh(1)^2 + 3*
x^2*cosh(1)^5 + 5*d*x*cosh(1)^4 - 2*(2*c^2*d^2*x^2 - d^2)*cosh(1)^3 + (c^4*d^4*x^2 - 2*c^2*d^4)*cosh(1))*sinh(
1))*log(-c*x + sqrt(c^2*x^2 - 1)) + 2*(c^4*d^5*x - 2*c^2*d^2*x^2*cosh(1)^3 - 3*c^2*d^3*x*cosh(1)^2 + (c^4*d^4*
x^2 - c^2*d^4)*cosh(1))*sinh(1) + (c^3*d^4*x*cosh(1)^2 + c^3*d^5*cosh(1) - c*d^2*x*cosh(1)^4 - c*d^2*x*sinh(1)
^4 - c*d^3*cosh(1)^3 - (4*c*d^2*x*cosh(1) + c*d^3)*sinh(1)^3 + (c^3*d^4*x - 6*c*d^2*x*cosh(1)^2 - 3*c*d^3*cosh
(1))*sinh(1)^2 + (2*c^3*d^4*x*cosh(1) + c^3*d^5 - 4*c*d^2*x*cosh(1)^3 - 3*c*d^3*cosh(1)^2)*sinh(1))*sqrt(c^2*x
^2 - 1))/(2*c^4*d^7*x*cosh(1)^2 + c^4*d^8*cosh(1) - 4*c^2*d^5*x*cosh(1)^4 + d^2*x^2*cosh(1)^7 + d^2*x^2*sinh(1
)^7 + 2*d^3*x*cosh(1)^6 + (7*d^2*x^2*cosh(1) + 2*d^3*x)*sinh(1)^6 - (2*c^2*d^4*x^2 - d^4)*cosh(1)^5 - (2*c^2*d
^4*x^2 - 21*d^2*x^2*cosh(1)^2 - 12*d^3*x*cosh(1) - d^4)*sinh(1)^5 - (4*c^2*d^5*x - 35*d^2*x^2*cosh(1)^3 - 30*d
^3*x*cosh(1)^2 + 5*(2*c^2*d^4*x^2 - d^4)*cosh(1))*sinh(1)^4 + (c^4*d^6*x^2 - 2*c^2*d^6)*cosh(1)^3 + (c^4*d^6*x
^2 - 16*c^2*d^5*x*cosh(1) - 2*c^2*d^6 + 35*d^2*x^2*cosh(1)^4 + 40*d^3*x*cosh(1)^3 - 10*(2*c^2*d^4*x^2 - d^4)*c
osh(1)^2)*sinh(1)^3 + (2*c^4*d^7*x - 24*c^2*d^5*x*cosh(1)^2 + 21*d^2*x^2*cosh(1)^5 + 30*d^3*x*cosh(1)^4 - 10*(
2*c^2*d^4*x^2 - d^4)*cosh(1)^3 + 3*(c^4*d^6*x^2 - 2*c^2*d^6)*cosh(1))*sinh(1)^2 + (4*c^4*d^7*x*cosh(1) + c^4*d
^8 - 16*c^2*d^5*x*cosh(1)^3 + 7*d^2*x^2*cosh(1)^6 + 12*d^3*x*cosh(1)^5 - 5*(2*c^2*d^4*x^2 - d^4)*cosh(1)^4 + 3
*(c^4*d^6*x^2 - 2*c^2*d^6)*cosh(1)^2)*sinh(1)), -1/2*(2*c^4*d^5*x*cosh(1) + c^4*d^6 - c^2*d^2*x^2*cosh(1)^4 -
c^2*d^2*x^2*sinh(1)^4 - 2*c^2*d^3*x*cosh(1)^3 - 2*(2*c^2*d^2*x^2*cosh(1) + c^2*d^3*x)*sinh(1)^3 + (c^4*d^4*x^2
- c^2*d^4)*cosh(1)^2 + (c^4*d^4*x^2 - 6*c^2*d^2*x^2*cosh(1)^2 - 6*c^2*d^3*x*cosh(1) - c^2*d^4)*sinh(1)^2 + 2*
(c^3*d^3*x^2*cosh(1)^2 + c^3*d^3*x^2*sinh(1)^2 + 2*c^3*d^4*x*cosh(1) + c^3*d^5 + 2*(c^3*d^3*x^2*cosh(1) + c^3*
d^4*x)*sinh(1))*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^4*d^4*x^2*cosh(1)^2 + 2*c^4*d^5*x*cosh(1) - 2*c^2*d^2*x
^2*cosh(1)^4 - 4*c^2*d^3*x*cosh(1)^3 + x^2*cosh(1)^6 + x^2*sinh(1)^6 + 2*d*x*cosh(1)^5 + 2*(3*x^2*cosh(1) + d*
x)*sinh(1)^5 - (2*c^2*d^2*x^2 - 15*x^2*cosh(1)^2 - 10*d*x*cosh(1))*sinh(1)^4 - 4*(2*c^2*d^2*x^2*cosh(1) + c^2*
d^3*x - 5*x^2*cosh(1)^3 - 5*d*x*cosh(1)^2)*sinh(1)^3 + (c^4*d^4*x^2 - 12*c^2*d^2*x^2*cosh(1)^2 - 12*c^2*d^3*x*
cosh(1) + 15*x^2*cosh(1)^4 + 20*d*x*cosh(1)^3)*sinh(1)^2 + 2*(c^4*d^4*x^2*cosh(1) + c^4*d^5*x - 4*c^2*d^2*x^2*
cosh(1)^3 - 6*c^2*d^3*x*cosh(1)^2 + 3*x^2*cosh(1)^5 + 5*d*x*cosh(1)^4)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) -
(2*c^4*d^5*x*cosh(1) + c^4*d^6 - 4*c^2*d^3*x*cosh(1)^3 + x^2*cosh(1)^6 + x^2*sinh(1)^6 + 2*d*x*cosh(1)^5 + 2*
(3*x^2*cosh(1) + d*x)*sinh(1)^5 - (2*c^2*d^2*x^...

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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 )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*x)/(e*x+d)**3,x)

[Out]

Integral(acosh(c*x)/(d + e*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, integration of abs
or sign ass

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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 )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)/(d + e*x)^3,x)

[Out]

int(acosh(c*x)/(d + e*x)^3, x)

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