3.120 \(\int \frac{(a+b \cosh ^{-1}(c+d x))^3}{(c e+d e x)^3} \, dx\)

Optimal. Leaf size=164 \[ \frac{3 b^3 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}-\frac{3 b^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac{3 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2} \]

[Out]

(-3*b*(a + b*ArcCosh[c + d*x])^2)/(2*d*e^3) + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x
])^2)/(2*d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^3/(2*d*e^3*(c + d*x)^2) - (3*b^2*(a + b*ArcCosh[c + d*x])
*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e^3) + (3*b^3*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(2*d*e^3)

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Rubi [A]  time = 0.369211, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5866, 12, 5662, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac{3 b^3 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}-\frac{3 b^2 \log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac{3 b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^3,x]

[Out]

(3*b*(a + b*ArcCosh[c + d*x])^2)/(2*d*e^3) + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]
)^2)/(2*d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^3/(2*d*e^3*(c + d*x)^2) - (3*b^2*(a + b*ArcCosh[c + d*x])*
Log[1 + E^(2*ArcCosh[c + d*x])])/(d*e^3) - (3*b^3*PolyLog[2, -E^(2*ArcCosh[c + d*x])])/(2*d*e^3)

Rule 5866

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

Rule 12

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

Rule 5662

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

Rule 5724

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh
[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} x^2 \sqrt{1+x}} \, dx,x,c+d x\right )}{2 d e^3}\\ &=\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}\\ &=\frac{3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac{3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac{3 b^3 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}\\ \end{align*}

Mathematica [A]  time = 1.04348, size = 266, normalized size = 1.62 \[ \frac{3 b^3 \left (\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )+\cosh ^{-1}(c+d x) \left (\frac{\sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}-\cosh ^{-1}(c+d x)-2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )\right )\right )+\frac{3 a^2 b \left (\sqrt{\frac{c+d x-1}{c+d x+1}} \left (c^2+2 c d x+c+d x (d x+1)\right )-\cosh ^{-1}(c+d x)\right )}{(c+d x)^2}-\frac{a^3}{(c+d x)^2}+6 a b^2 \left (-\log (c+d x)-\frac{\cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}+\frac{\sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}\right )-\frac{b^3 \cosh ^{-1}(c+d x)^3}{(c+d x)^2}}{2 d e^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^3,x]

[Out]

(-(a^3/(c + d*x)^2) + (3*a^2*b*(Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(c + c^2 + 2*c*d*x + d*x*(1 + d*x)) - ArcCo
sh[c + d*x]))/(c + d*x)^2 - (b^3*ArcCosh[c + d*x]^3)/(c + d*x)^2 + 6*a*b^2*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)
]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x) - ArcCosh[c + d*x]^2/(2*(c + d*x)^2) - Log[c + d*x]) + 3*b^3*(ArcC
osh[c + d*x]*(-ArcCosh[c + d*x] + (Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x
) - 2*Log[1 + E^(-2*ArcCosh[c + d*x])]) + PolyLog[2, -E^(-2*ArcCosh[c + d*x])]))/(2*d*e^3)

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Maple [B]  time = 0.067, size = 375, normalized size = 2.3 \begin{align*} -{\frac{{a}^{3}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{3\,{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{2\,d{e}^{3} \left ( dx+c \right ) }\sqrt{dx+c-1}\sqrt{dx+c+1}}+{\frac{3\,{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{2\,d{e}^{3}}}-{\frac{{b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}-3\,{\frac{{b}^{3}{\rm arccosh} \left (dx+c\right )\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{d{e}^{3}}}-{\frac{3\,{b}^{3}}{2\,d{e}^{3}}{\it polylog} \left ( 2,- \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2} \right ) }+3\,{\frac{a{b}^{2}{\rm arccosh} \left (dx+c\right )}{d{e}^{3}}}+3\,{\frac{a{b}^{2}{\rm arccosh} \left (dx+c\right )\sqrt{dx+c+1}\sqrt{dx+c-1}}{d{e}^{3} \left ( dx+c \right ) }}-{\frac{3\,a{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}-3\,{\frac{a{b}^{2}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }{d{e}^{3}}}-{\frac{3\,{a}^{2}b{\rm arccosh} \left (dx+c\right )}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{3\,{a}^{2}b}{2\,d{e}^{3} \left ( dx+c \right ) }\sqrt{dx+c-1}\sqrt{dx+c+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^3,x)

[Out]

-1/2/d*a^3/e^3/(d*x+c)^2+3/2/d*b^3/e^3*arccosh(d*x+c)^2/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)+3/2/d*b^3/e^3*
arccosh(d*x+c)^2-1/2/d*b^3/e^3*arccosh(d*x+c)^3/(d*x+c)^2-3/d*b^3/e^3*arccosh(d*x+c)*ln((d*x+c+(d*x+c-1)^(1/2)
*(d*x+c+1)^(1/2))^2+1)-3/2/d*b^3/e^3*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/d*a*b^2/e^3*arcco
sh(d*x+c)+3/d*a*b^2/e^3*arccosh(d*x+c)/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)-3/2/d*a*b^2/e^3*arccosh(d*x+c)^
2/(d*x+c)^2-3/d*a*b^2/e^3*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2+1)-3/2/d*a^2*b/e^3/(d*x+c)^2*arccosh(d*
x+c)+3/2/d*a^2*b/e^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)

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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((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (d x + c\right ) + a^{3}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*arccosh(d*x + c)^3 + 3*a*b^2*arccosh(d*x + c)^2 + 3*a^2*b*arccosh(d*x + c) + a^3)/(d^3*e^3*x^3 +
3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{3 a^{2} b \operatorname{acosh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(b**3*acosh(c + d*x)**3/(c**3 + 3
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(3*a*b**2*acosh(c + d*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2
*x**2 + d**3*x**3), x) + Integral(3*a**2*b*acosh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))
/e**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(d*x + c) + a)^3/(d*e*x + c*e)^3, x)