3.119 \(\int \frac{(a+b \cosh ^{-1}(c+d x))^3}{(c e+d e x)^2} \, dx\)

Optimal. Leaf size=186 \[ -\frac{6 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{6 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2} \]

[Out]

-((a + b*ArcCosh[c + d*x])^3/(d*e^2*(c + d*x))) + (6*b*(a + b*ArcCosh[c + d*x])^2*ArcTan[E^ArcCosh[c + d*x]])/
(d*e^2) - ((6*I)*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((6*I)*b^2*(a + b
*ArcCosh[c + d*x])*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2) + ((6*I)*b^3*PolyLog[3, (-I)*E^ArcCosh[c + d*x]])
/(d*e^2) - ((6*I)*b^3*PolyLog[3, I*E^ArcCosh[c + d*x]])/(d*e^2)

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Rubi [A]  time = 0.36201, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5866, 12, 5662, 5761, 4180, 2531, 2282, 6589} \[ -\frac{6 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 i b^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 i b^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{6 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^2,x]

[Out]

-((a + b*ArcCosh[c + d*x])^3/(d*e^2*(c + d*x))) + (6*b*(a + b*ArcCosh[c + d*x])^2*ArcTan[E^ArcCosh[c + d*x]])/
(d*e^2) - ((6*I)*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((6*I)*b^2*(a + b
*ArcCosh[c + d*x])*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2) + ((6*I)*b^3*PolyLog[3, (-I)*E^ArcCosh[c + d*x]])
/(d*e^2) - ((6*I)*b^3*PolyLog[3, I*E^ArcCosh[c + d*x]])/(d*e^2)

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 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 4180

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{6 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{6 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}-\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{6 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{6 b \left (a+b \cosh ^{-1}(c+d x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 i b^3 \text{Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 i b^3 \text{Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ \end{align*}

Mathematica [A]  time = 1.09634, size = 327, normalized size = 1.76 \[ -\frac{3 i a b^2 \left (2 \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )+\cosh ^{-1}(c+d x) \left (-\frac{i \cosh ^{-1}(c+d x)}{c+d x}+2 \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )\right )+b^3 \left (\frac{\cosh ^{-1}(c+d x)^3}{c+d x}-3 i \left (-2 \cosh ^{-1}(c+d x) \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right )-2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c+d x)}\right )+\cosh ^{-1}(c+d x)^2 \left (-\left (\log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )\right )\right )\right )+3 a^2 b \tan ^{-1}\left (\frac{1}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )+\frac{3 a^2 b \cosh ^{-1}(c+d x)}{c+d x}+\frac{a^3}{c+d x}}{d e^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^2,x]

[Out]

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

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{3}}{ \left ( dex+ce \right ) ^{2}}}\, dx \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)^2,x)

[Out]

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

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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)^2,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^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, 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)^2,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^2*e^2*x^2 +
2*c*d*e^2*x + c^2*e^2), x)

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

[Out]

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

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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((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError