3.7 \(\int \frac{\cosh ^{-1}(c x)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=195 \[ -\frac{c \sqrt{c x-1} \sqrt{c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac{c^3 d \sqrt{c x-1} \sqrt{c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3} \]

[Out]

-(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2) - (c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c
*d - e)^2*(c*d + e)^2*(d + e*x)) - ArcCosh[c*x]/(3*e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d +
e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2))

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Rubi [A]  time = 0.237232, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5802, 103, 151, 12, 93, 208} \[ -\frac{c \sqrt{c x-1} \sqrt{c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac{c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac{c^3 d \sqrt{c x-1} \sqrt{c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*x]/(d + e*x)^4,x]

[Out]

-(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2) - (c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c
*d - e)^2*(c*d + e)^2*(d + e*x)) - ArcCosh[c*x]/(3*e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d +
e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2))

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]

Rule 103

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

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

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]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac{c \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)^3} \, dx}{3 e}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac{c \int \frac{-2 c^2 d+c^2 e x}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2} \, dx}{6 e \left (c^2 d^2-e^2\right )}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{c^3 d \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac{c \int \frac{c^2 \left (2 c^2 d^2+e^2\right )}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{c^3 d \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac{\left (c^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{c^3 d \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac{\left (c^3 \left (2 c^2 d^2+e^2\right )\right ) \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 )}{3 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac{c \sqrt{-1+c x} \sqrt{1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac{c^3 d \sqrt{-1+c x} \sqrt{1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac{\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac{c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c d+e} \sqrt{1+c x}}{\sqrt{c d-e} \sqrt{-1+c x}}\right )}{3 (c d-e)^{5/2} e (c d+e)^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.550665, size = 244, normalized size = 1.25 \[ \frac{1}{6} \left (\frac{c \sqrt{c x-1} \sqrt{c x+1} \left (e^2-c^2 d (4 d+3 e x)\right )}{\left (e^2-c^2 d^2\right )^2 (d+e x)^2}-\frac{i c^3 \left (2 c^2 d^2+e^2\right ) \log \left (\frac{12 e^2 (e-c d)^2 (c d+e)^2 \left (\sqrt{c x-1} \sqrt{c x+1} \sqrt{e^2-c^2 d^2}-i c^2 d x-i e\right )}{c^3 \sqrt{e^2-c^2 d^2} \left (2 c^2 d^2+e^2\right ) (d+e x)}\right )}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{2 \cosh ^{-1}(c x)}{e (d+e x)^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[c*x]/(d + e*x)^4,x]

[Out]

((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2 - c^2*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcCosh[
c*x])/(e*(d + e*x)^3) - (I*c^3*(2*c^2*d^2 + e^2)*Log[(12*e^2*(-(c*d) + e)^2*(c*d + e)^2*((-I)*e - I*c^2*d*x +
Sqrt[-(c^2*d^2) + e^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*Sqrt[-(c^2*d^2) + e^2]*(2*c^2*d^2 + e^2)*(d + e*x))
])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/6

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Maple [B]  time = 0.023, size = 1108, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(c*x)/(e*x+d)^4,x)

[Out]

-1/3*c^3/(c*e*x+c*d)^3/e*arccosh(c*x)-1/3*c^7*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/((
c^2*d^2-e^2)/e^2)^(1/2)/(c^2*d^2-e^2)/(c*e*x+c*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))*x^2*d^2-2/3*c^7/e*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/((c^2*d^2-e
^2)/e^2)^(1/2)/(c^2*d^2-e^2)/(c*e*x+c*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))*x*d^3-1/6*c^5*e^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^2*d^2-e^2)/e^2)^
(1/2)/(c^2*d^2-e^2)/(c*e*x+c*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))
*x^2-1/3*c^7/e^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^2*d^2-e^2)/e^2)^(1/2)/(c^2*
d^2-e^2)/(c*e*x+c*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))*d^4-1/2*c^
5*e*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*x*d-1/3*c^5*e*(c*x+1)^(1/2)*(c*x-1
)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/((c^2*d^2-e^2)/e^2)^(1/2)/(c^2*d^2-e^2)/(c*e*x+c*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))*x*d-2/3*c^5*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*d+e)
/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*d^2-1/6*c^5*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)
/((c^2*d^2-e^2)/e^2)^(1/2)/(c^2*d^2-e^2)/(c*e*x+c*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))*d^2+1/6*c^3*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.39307, size = 3479, normalized size = 17.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/6*(3*c^6*d^9 - 3*c^4*d^7*e^2 + 3*(c^6*d^6*e^3 - c^4*d^4*e^5)*x^3 + 9*(c^6*d^7*e^2 - c^4*d^5*e^4)*x^2 - (2*
c^5*d^8 + c^3*d^6*e^2 + (2*c^5*d^5*e^3 + c^3*d^3*e^5)*x^3 + 3*(2*c^5*d^6*e^2 + c^3*d^4*e^4)*x^2 + 3*(2*c^5*d^7
*e + c^3*d^5*e^3)*x)*sqrt(c^2*d^2 - e^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)) + 9*(c^6*d^8*e - c^4*d^6*e^3)*x - 2*((c^6*d^6*
e^3 - 3*c^4*d^4*e^5 + 3*c^2*d^2*e^7 - e^9)*x^3 + 3*(c^6*d^7*e^2 - 3*c^4*d^5*e^4 + 3*c^2*d^3*e^6 - d*e^8)*x^2 +
3*(c^6*d^8*e - 3*c^4*d^6*e^3 + 3*c^2*d^4*e^5 - d^2*e^7)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(c^6*d^9 - 3*c^4*
d^7*e^2 + 3*c^2*d^5*e^4 - d^3*e^6 + (c^6*d^6*e^3 - 3*c^4*d^4*e^5 + 3*c^2*d^2*e^7 - e^9)*x^3 + 3*(c^6*d^7*e^2 -
3*c^4*d^5*e^4 + 3*c^2*d^3*e^6 - d*e^8)*x^2 + 3*(c^6*d^8*e - 3*c^4*d^6*e^3 + 3*c^2*d^4*e^5 - d^2*e^7)*x)*log(-
c*x + sqrt(c^2*x^2 - 1)) + (4*c^5*d^8*e - 5*c^3*d^6*e^3 + c*d^4*e^5 + 3*(c^5*d^6*e^3 - c^3*d^4*e^5)*x^2 + (7*c
^5*d^7*e^2 - 8*c^3*d^5*e^4 + c*d^3*e^6)*x)*sqrt(c^2*x^2 - 1))/(c^6*d^12*e - 3*c^4*d^10*e^3 + 3*c^2*d^8*e^5 - d
^6*e^7 + (c^6*d^9*e^4 - 3*c^4*d^7*e^6 + 3*c^2*d^5*e^8 - d^3*e^10)*x^3 + 3*(c^6*d^10*e^3 - 3*c^4*d^8*e^5 + 3*c^
2*d^6*e^7 - d^4*e^9)*x^2 + 3*(c^6*d^11*e^2 - 3*c^4*d^9*e^4 + 3*c^2*d^7*e^6 - d^5*e^8)*x), -1/6*(3*c^6*d^9 - 3*
c^4*d^7*e^2 + 3*(c^6*d^6*e^3 - c^4*d^4*e^5)*x^3 + 9*(c^6*d^7*e^2 - c^4*d^5*e^4)*x^2 + 2*(2*c^5*d^8 + c^3*d^6*e
^2 + (2*c^5*d^5*e^3 + c^3*d^3*e^5)*x^3 + 3*(2*c^5*d^6*e^2 + c^3*d^4*e^4)*x^2 + 3*(2*c^5*d^7*e + c^3*d^5*e^3)*x
)*sqrt(-c^2*d^2 + e^2)*arctan(-(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)) + 9*(c^6*d^8*e - c^4*d^6*e^3)*x - 2*((c^6*d^6*e^3 - 3*c^4*d^4*e^5 + 3*c^2*d^2*e^7 - e^9)*x^3
+ 3*(c^6*d^7*e^2 - 3*c^4*d^5*e^4 + 3*c^2*d^3*e^6 - d*e^8)*x^2 + 3*(c^6*d^8*e - 3*c^4*d^6*e^3 + 3*c^2*d^4*e^5
- d^2*e^7)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(c^6*d^9 - 3*c^4*d^7*e^2 + 3*c^2*d^5*e^4 - d^3*e^6 + (c^6*d^6*e
^3 - 3*c^4*d^4*e^5 + 3*c^2*d^2*e^7 - e^9)*x^3 + 3*(c^6*d^7*e^2 - 3*c^4*d^5*e^4 + 3*c^2*d^3*e^6 - d*e^8)*x^2 +
3*(c^6*d^8*e - 3*c^4*d^6*e^3 + 3*c^2*d^4*e^5 - d^2*e^7)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (4*c^5*d^8*e - 5*c^
3*d^6*e^3 + c*d^4*e^5 + 3*(c^5*d^6*e^3 - c^3*d^4*e^5)*x^2 + (7*c^5*d^7*e^2 - 8*c^3*d^5*e^4 + c*d^3*e^6)*x)*sqr
t(c^2*x^2 - 1))/(c^6*d^12*e - 3*c^4*d^10*e^3 + 3*c^2*d^8*e^5 - d^6*e^7 + (c^6*d^9*e^4 - 3*c^4*d^7*e^6 + 3*c^2*
d^5*e^8 - d^3*e^10)*x^3 + 3*(c^6*d^10*e^3 - 3*c^4*d^8*e^5 + 3*c^2*d^6*e^7 - d^4*e^9)*x^2 + 3*(c^6*d^11*e^2 - 3
*c^4*d^9*e^4 + 3*c^2*d^7*e^6 - d^5*e^8)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*x)/(e*x+d)**4,x)

[Out]

Integral(acosh(c*x)/(d + e*x)**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(c*x)/(e*x+d)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError