3.1.11 \(\int \frac {\cosh ^{-1}(c x)^2}{d+e x} \, dx\) [11]

Optimal. Leaf size=272 \[ -\frac {\cosh ^{-1}(c x)^3}{3 e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {PolyLog}\left (3,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {PolyLog}\left (3,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \]

[Out]

-1/3*arccosh(c*x)^3/e+arccosh(c*x)^2*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e+arc
cosh(c*x)^2*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e+2*arccosh(c*x)*polylog(2,-e*
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e+2*arccosh(c*x)*polylog(2,-e*(c*x+(c*x-1)^(1/2)*
(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e-2*polylog(3,-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2
)^(1/2)))/e-2*polylog(3,-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e

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Rubi [A]
time = 0.30, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5962, 5681, 2221, 2611, 2320, 6724} \begin {gather*} \frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Li}_3\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Li}_3\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\cosh ^{-1}(c x)^3}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*x]^2/(d + e*x),x]

[Out]

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 5681

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 5962

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

Rule 6724

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 {\cosh ^{-1}(c x)^2}{d+e x} \, dx &=\text {Subst}\left (\int \frac {x^2 \sinh (x)}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac {\cosh ^{-1}(c x)^3}{3 e}+\text {Subst}\left (\int \frac {e^x x^2}{c d-\sqrt {c^2 d^2-e^2}+e e^x} \, dx,x,\cosh ^{-1}(c x)\right )+\text {Subst}\left (\int \frac {e^x x^2}{c d+\sqrt {c^2 d^2-e^2}+e e^x} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac {\cosh ^{-1}(c x)^3}{3 e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Subst}\left (\int x \log \left (1+\frac {e e^x}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}-\frac {2 \text {Subst}\left (\int x \log \left (1+\frac {e e^x}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\cosh ^{-1}(c x)^3}{3 e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Subst}\left (\int \text {Li}_2\left (-\frac {e e^x}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}-\frac {2 \text {Subst}\left (\int \text {Li}_2\left (-\frac {e e^x}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\cosh ^{-1}(c x)^3}{3 e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {e x}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {e x}{c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e}\\ &=-\frac {\cosh ^{-1}(c x)^3}{3 e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 \cosh ^{-1}(c x) \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Li}_3\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 \text {Li}_3\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 252, normalized size = 0.93 \begin {gather*} -\frac {\cosh ^{-1}(c x)^3-3 \cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-3 \cosh ^{-1}(c x)^2 \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-6 \cosh ^{-1}(c x) \text {PolyLog}\left (2,\frac {e e^{\cosh ^{-1}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-6 \cosh ^{-1}(c x) \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+6 \text {PolyLog}\left (3,\frac {e e^{\cosh ^{-1}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+6 \text {PolyLog}\left (3,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[c*x]^2/(d + e*x),x]

[Out]

-1/3*(ArcCosh[c*x]^3 - 3*ArcCosh[c*x]^2*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])] - 3*ArcCosh[c*
x]^2*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])] - 6*ArcCosh[c*x]*PolyLog[2, (e*E^ArcCosh[c*x])/(-
(c*d) + Sqrt[c^2*d^2 - e^2])] - 6*ArcCosh[c*x]*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))] +
6*PolyLog[3, (e*E^ArcCosh[c*x])/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 6*PolyLog[3, -((e*E^ArcCosh[c*x])/(c*d + Sq
rt[c^2*d^2 - e^2]))])/e

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Maple [F]
time = 1.46, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (c x \right )^{2}}{e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(c*x)^2/(x*e + d), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccosh(c*x)^2/(x*e + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*x)**2/(e*x+d),x)

[Out]

Integral(acosh(c*x)**2/(d + e*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(c*x)^2/(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(c*x)^2/(d + e*x),x)

[Out]

int(acosh(c*x)^2/(d + e*x), x)

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