3.1.13 \(\int \frac {\cosh ^{-1}(c x)^2}{(d+e x)^3} \, dx\) [13]

Optimal. Leaf size=352 \[ -\frac {c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.48, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5963, 5980, 3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \frac {c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {c^3 d \cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {c^3 d \cosh ^{-1}(c x) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\cosh ^{-1}(c x)^2}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[c*x]^2/(d + e*x)^3,x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

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 2438

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]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3401

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

Rule 3405

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

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]

Rule 5980

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(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*(c*f + g*Cosh[x])^m,
x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2
, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 3.13, size = 936, normalized size = 2.66 \begin {gather*} c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{(c d-e) (c d+e) (c d+c e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (c d+c e x)^2}+\frac {\log \left (1+\frac {e x}{d}\right )}{c^2 d^2 e-e^3}+\frac {c d \left (2 \cosh ^{-1}(c x) \text {ArcTan}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )-2 i \text {ArcCos}\left (-\frac {c d}{e}\right ) \text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\left (\text {ArcCos}\left (-\frac {c d}{e}\right )+2 \left (\text {ArcTan}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{-\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )+\left (\text {ArcCos}\left (-\frac {c d}{e}\right )-2 \left (\text {ArcTan}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c d+c e x}}\right )-\left (\text {ArcCos}\left (-\frac {c d}{e}\right )+2 \text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {-c^2 d^2+e^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {c d}{e}\right )-2 \text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {-c^2 d^2+e^2}\right ) \left (1+\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (c d-i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (c d+i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{e \left (-c^2 d^2+e^2\right )^{3/2}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/((c*d - e)*(c*d + e)*(c*d + c*e*x))) - ArcCosh[c*x]
^2/(2*e*(c*d + c*e*x)^2) + Log[1 + (e*x)/d]/(c^2*d^2*e - e^3) + (c*d*(2*ArcCosh[c*x]*ArcTan[((c*d + e)*Coth[Ar
cCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] - (2*I)*ArcCos[-((c*d)/e)]*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/S
qrt[-(c^2*d^2) + e^2]] + (ArcCos[-((c*d)/e)] + 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^
2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]]))*Log[Sqrt[-(c^2*d^2) + e^2]/(Sqrt[2]
*Sqrt[e]*E^(ArcCosh[c*x]/2)*Sqrt[c*d + c*e*x])] + (ArcCos[-((c*d)/e)] - 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]
/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]]))*Log[(Sqrt[
-(c^2*d^2) + e^2]*E^(ArcCosh[c*x]/2))/(Sqrt[2]*Sqrt[e]*Sqrt[c*d + c*e*x])] - (ArcCos[-((c*d)/e)] + 2*ArcTan[((
-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]])*Log[((c*d + e)*(c*d - e + I*Sqrt[-(c^2*d^2) + e^2])
*(-1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] - (ArcCos[-((c*d)
/e)] - 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]])*Log[((c*d + e)*(-(c*d) + e + I*Sq
rt[-(c^2*d^2) + e^2])*(1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2])
)] + I*(PolyLog[2, ((c*d - I*Sqrt[-(c^2*d^2) + e^2])*(c*d + e - I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2])
)/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] - PolyLog[2, ((c*d + I*Sqrt[-(c^2*d^2) + e^2]
)*(c*d + e - I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCo
sh[c*x]/2]))])))/(e*(-(c^2*d^2) + e^2)^(3/2)))

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Maple [A]
time = 7.63, size = 605, normalized size = 1.72

method	result	size
derivativedivides	\(\frac {-\frac {c^{3} \mathrm {arccosh}\left (c x \right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, c d e +2 \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} c x +c^{2} d^{2} \mathrm {arccosh}\left (c x \right )-2 c^{2} d^{2}-4 c^{2} d e x -2 e^{2} c^{2} x^{2}-e^{2} \mathrm {arccosh}\left (c x \right )\right )}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2}}+\frac {c^{4} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} \dilog \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} \dilog \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{3} \ln \left (2 d c \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}}{c}\)	\(605\)
default	\(\frac {-\frac {c^{3} \mathrm {arccosh}\left (c x \right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, c d e +2 \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} c x +c^{2} d^{2} \mathrm {arccosh}\left (c x \right )-2 c^{2} d^{2}-4 c^{2} d e x -2 e^{2} c^{2} x^{2}-e^{2} \mathrm {arccosh}\left (c x \right )\right )}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (e c x +c d \right )^{2}}+\frac {c^{4} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} \dilog \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} \dilog \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{3} \ln \left (2 d c \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}}{c}\)	\(605\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(c*x)^2/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/2*c^3*arccosh(c*x)*(2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*d*e+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*c*x+c^2*d^2*
arccosh(c*x)-2*c^2*d^2-4*c^2*d*e*x-2*e^2*c^2*x^2-e^2*arccosh(c*x))/e/(c^2*d^2-e^2)/(c*e*x+c*d)^2+1/(c^2*d^2-e^
2)^(3/2)/e*c^4*arccosh(c*x)*ln((-c*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e
^2)^(1/2)))*d-1/(c^2*d^2-e^2)^(3/2)/e*c^4*arccosh(c*x)*ln((c*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^
2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))*d+1/(c^2*d^2-e^2)^(3/2)/e*c^4*dilog((-c*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1
/2))+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2)))*d-1/(c^2*d^2-e^2)^(3/2)/e*c^4*dilog((c*d+e*(c*x+(c*x-1)^
(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))*d+1/(c^2*d^2-e^2)/e*c^3*ln(2*d*c*(c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+e)-2/(c^2*d^2-e^2)/e*c^3*ln(c*x+(c*x-1)^(1/2)*
(c*x+1)^(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)^2/(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 [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)^3,x, algorithm="fricas")

[Out]

integral(arccosh(c*x)^2/(x^3*e^3 + 3*d*x^2*e^2 + 3*d^2*x*e + d^3), 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 )}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(c*x)**2/(e*x+d)**3,x)

[Out]

Integral(acosh(c*x)**2/(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)^2/(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 = -0.00 \begin {gather*} \int \frac {{\mathrm {acosh}\left (c\,x\right )}^2}{{\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)^2/(d + e*x)^3,x)

[Out]

int(acosh(c*x)^2/(d + e*x)^3, x)

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