3.1.25 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{(d+e x)^2} \, dx\) [25]

Optimal. Leaf size=279 \[ -\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]

[Out]

-(a+b*arccosh(c*x))^2/e/(e*x+d)+2*b*c*(a+b*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)^(1/2)-2*b*c*(a+b*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)^(1/2)+2*b^2*c*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)^(1/2)-2*b^2*c*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)^(1/2)

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Rubi [A]
time = 0.43, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5963, 5980, 3401, 2296, 2221, 2317, 2438} \begin {gather*} \frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 {\left (a+b \cosh ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \text {Subst}\left (\int \frac {a+b x}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{e+2 c d e^x+e e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d-2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 d^2-e^2}}-\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d+2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c\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 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.76, size = 950, normalized size = 3.41 \begin {gather*} -\frac {\frac {a^2}{d+e x}+2 a b c \left (\frac {\cosh ^{-1}(c x)}{c d+c e x}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {c d-e} \sqrt {\frac {-1+c x}{1+c x}}}{\sqrt {-c d-e}}\right )}{\sqrt {-c d-e} \sqrt {c d-e}}\right )+b^2 c \left (\frac {\cosh ^{-1}(c x)^2}{c d+c e x}+\frac {2 \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+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+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 )}{\sqrt {-c^2 d^2+e^2}}\right )}{e} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a^2/(d + e*x) + 2*a*b*c*(ArcCosh[c*x]/(c*d + c*e*x) + (2*ArcTan[(Sqrt[c*d - e]*Sqrt[(-1 + c*x)/(1 + c*x)])/
Sqrt[-(c*d) - e]])/(Sqrt[-(c*d) - e]*Sqrt[c*d - e])) + b^2*c*(ArcCosh[c*x]^2/(c*d + c*e*x) + (2*(2*ArcCosh[c*x
]*ArcTan[((c*d + e)*Coth[ArcCosh[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])/Sqrt[-(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 + 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 + e*x)])] - (ArcCo
s[-((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*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]))] + 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[ArcCosh[c*x]/2]))])))/Sqrt[-(c^2*d^2) + e^2]))/e)

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Maple [A]
time = 9.34, size = 556, normalized size = 1.99

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} c^{2}}{\left (e c x +c d \right ) e}-\frac {b^{2} c^{2} \mathrm {arccosh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 b^{2} c^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 b^{2} c^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 b^{2} c^{2} \dilog \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 b^{2} c^{2} \dilog \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 a b \,c^{2} \mathrm {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {2 a b \,c^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}}{c}\)	\(556\)
default	\(\frac {-\frac {a^{2} c^{2}}{\left (e c x +c d \right ) e}-\frac {b^{2} c^{2} \mathrm {arccosh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 b^{2} c^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 b^{2} c^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 b^{2} c^{2} \dilog \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 b^{2} c^{2} \dilog \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 a b \,c^{2} \mathrm {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {2 a b \,c^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}}{c}\)	\(556\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-a^2*c^2/(c*e*x+c*d)/e-b^2*c^2*arccosh(c*x)^2/e/(c*e*x+c*d)+2*b^2*c^2/e*arccosh(c*x)/(c^2*d^2-e^2)^(1/2)*
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)))-2*b^2*c^2/e*arcc
osh(c*x)/(c^2*d^2-e^2)^(1/2)*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)))+2*b^2*c^2/e/(c^2*d^2-e^2)^(1/2)*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)))-2*b^2*c^2/e/(c^2*d^2-e^2)^(1/2)*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)))-2*a*b*c^2/(c*e*x+c*d)/e*arccosh(c*x)-2*a*b*c^2/e^2*(c*x+1)^(1/2
)*(c*x-1)^(1/2)*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))/((c^2*d^2-e^2)/e^
2)^(1/2)/(c^2*x^2-1)^(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((a+b*arccosh(c*x))^2/(e*x+d)^2,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((a+b*arccosh(c*x))^2/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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