\(\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx\) [45]

Optimal result

Integrand size = 23, antiderivative size = 264 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=-\frac {(a+b \text {arccosh}(c+d x))^4}{d e^2 (c+d x)}+\frac {8 b (a+b \text {arccosh}(c+d x))^3 \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{d e^2}-\frac {12 i b^2 (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{d e^2}+\frac {12 i b^2 (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{d e^2}+\frac {24 i b^3 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )}{d e^2}-\frac {24 i b^3 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )}{d e^2}-\frac {24 i b^4 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(c+d x)}\right )}{d e^2}+\frac {24 i b^4 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(c+d x)}\right )}{d e^2} \] Output:

-(a+b*arccosh(d*x+c))^4/d/e^2/(d*x+c)+8*b*(a+b*arccosh(d*x+c))^3*arctan(d* 
x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/d/e^2-12*I*b^2*(a+b*arccosh(d*x+c))^2 
*polylog(2,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^2+12*I*b^2*(a+b 
*arccosh(d*x+c))^2*polylog(2,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/ 
e^2+24*I*b^3*(a+b*arccosh(d*x+c))*polylog(3,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x 
+c+1)^(1/2)))/d/e^2-24*I*b^3*(a+b*arccosh(d*x+c))*polylog(3,I*(d*x+c+(d*x+ 
c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^2-24*I*b^4*polylog(4,-I*(d*x+c+(d*x+c-1)^ 
(1/2)*(d*x+c+1)^(1/2)))/d/e^2+24*I*b^4*polylog(4,I*(d*x+c+(d*x+c-1)^(1/2)* 
(d*x+c+1)^(1/2)))/d/e^2
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(872\) vs. \(2(264)=528\).

Time = 1.77 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.30 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^2,x]
 

Output:

(-(a^4/(c + d*x)) + 4*a^3*b*(-(ArcCosh[c + d*x]/(c + d*x)) + 2*ArcTan[Tanh 
[ArcCosh[c + d*x]/2]]) - (6*I)*a^2*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^ArcCos 
h[c + d*x]]) + 2*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - 2*PolyLog[2, I/E^Ar 
cCosh[c + d*x]]) + 4*a*b^3*(-(ArcCosh[c + d*x]^3/(c + d*x)) + (3*I)*(-(Arc 
Cosh[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*Pol 
yLog[3, I/E^ArcCosh[c + d*x]])) + b^4*(((-7*I)/16)*Pi^4 + (Pi^3*ArcCosh[c 
+ d*x])/2 - ((3*I)/2)*Pi^2*ArcCosh[c + d*x]^2 - 2*Pi*ArcCosh[c + d*x]^3 + 
I*ArcCosh[c + d*x]^4 - ArcCosh[c + d*x]^4/(c + d*x) + (Pi^3*Log[1 + I/E^Ar 
cCosh[c + d*x]])/2 - (3*I)*Pi^2*ArcCosh[c + d*x]*Log[1 + I/E^ArcCosh[c + d 
*x]] - 6*Pi*ArcCosh[c + d*x]^2*Log[1 + I/E^ArcCosh[c + d*x]] + (4*I)*ArcCo 
sh[c + d*x]^3*Log[1 + I/E^ArcCosh[c + d*x]] + (3*I)*Pi^2*ArcCosh[c + d*x]* 
Log[1 - I*E^ArcCosh[c + d*x]] + 6*Pi*ArcCosh[c + d*x]^2*Log[1 - I*E^ArcCos 
h[c + d*x]] - (Pi^3*Log[1 + I*E^ArcCosh[c + d*x]])/2 - (4*I)*ArcCosh[c + d 
*x]^3*Log[1 + I*E^ArcCosh[c + d*x]] + (Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[c 
+ d*x])/4]])/2 + (3*I)*(Pi - (2*I)*ArcCosh[c + d*x])^2*PolyLog[2, (-I)/E^A 
rcCosh[c + d*x]] - (12*I)*ArcCosh[c + d*x]^2*PolyLog[2, (-I)*E^ArcCosh[c + 
 d*x]] + (3*I)*Pi^2*PolyLog[2, I*E^ArcCosh[c + d*x]] + 12*Pi*ArcCosh[c ...
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.82, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6411, 27, 6298, 6362, 3042, 4668, 3011, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{e^2 (c+d x)^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c+d x)^2}d(c+d x)}{d e^2}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {4 b \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}}{d e^2}\)

\(\Big \downarrow \) 6362

\(\displaystyle \frac {4 b \int \frac {(a+b \text {arccosh}(c+d x))^3}{c+d x}d\text {arccosh}(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}}{d e^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}+4 b \int (a+b \text {arccosh}(c+d x))^3 \csc \left (i \text {arccosh}(c+d x)+\frac {\pi }{2}\right )d\text {arccosh}(c+d x)}{d e^2}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}+4 b \left (-3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+3 i b \int (a+b \text {arccosh}(c+d x))^2 \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )}{d e^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}+4 b \left (3 i b \left (2 b \int (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \int (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )}{d e^2}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}+4 b \left (3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )}{d e^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}+4 b \left (3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int e^{-\text {arccosh}(c+d x)} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \int e^{-\text {arccosh}(c+d x)} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3\right )}{d e^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}+4 b \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^3+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-b \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))^2\right )\right )}{d e^2}\)

Input:

Int[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^2,x]
 

Output:

(-((a + b*ArcCosh[c + d*x])^4/(c + d*x)) + 4*b*(2*(a + b*ArcCosh[c + d*x]) 
^3*ArcTan[E^ArcCosh[c + d*x]] + (3*I)*b*(-((a + b*ArcCosh[c + d*x])^2*Poly 
Log[2, (-I)*E^ArcCosh[c + d*x]]) + 2*b*((a + b*ArcCosh[c + d*x])*PolyLog[3 
, (-I)*E^ArcCosh[c + d*x]] - b*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])) - (3* 
I)*b*(-((a + b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]]) + 2*b 
*((a + b*ArcCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*x]] - b*PolyLog[4, 
 I*E^ArcCosh[c + d*x]]))))/(d*e^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4668

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] + (-Simp[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] + Simp[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 6298

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

rule 6362

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 
_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ 
Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   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] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte 
gerQ[m]
 

rule 6411

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

rule 7163

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

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{2}}d x\]

Input:

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x)
 

Output:

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x)
 

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="fricas")
 

Output:

integral((b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2* 
arccosh(d*x + c)^2 + 4*a^3*b*arccosh(d*x + c) + a^4)/(d^2*e^2*x^2 + 2*c*d* 
e^2*x + c^2*e^2), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{4}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \] Input:

integrate((a+b*acosh(d*x+c))**4/(d*e*x+c*e)**2,x)
 

Output:

(Integral(a**4/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**4*acosh(c + 
d*x)**4/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(4*a*b**3*acosh(c + d*x 
)**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(6*a**2*b**2*acosh(c + d*x 
)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(4*a**3*b*acosh(c + d*x)/( 
c**2 + 2*c*d*x + d**2*x**2), x))/e**2
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \] Input:

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^2,x)
                                                                                    
                                                                                    
 

Output:

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^2, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^2} \, dx=\frac {4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a^{3} b \,c^{2}+4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a^{3} b c d x +\left (\int \frac {\mathit {acosh} \left (d x +c \right )^{4}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b^{4} c^{2}+\left (\int \frac {\mathit {acosh} \left (d x +c \right )^{4}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b^{4} c d x +4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{3}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a \,b^{3} c^{2}+4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{3}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a \,b^{3} c d x +6 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a^{2} b^{2} c^{2}+6 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a^{2} b^{2} c d x +a^{4} x}{c \,e^{2} \left (d x +c \right )} \] Input:

int((a+b*acosh(d*x+c))^4/(d*e*x+c*e)^2,x)
 

Output:

(4*int(acosh(c + d*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**3*b*c**2 + 4*int( 
acosh(c + d*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a**3*b*c*d*x + int(acosh(c 
+ d*x)**4/(c**2 + 2*c*d*x + d**2*x**2),x)*b**4*c**2 + int(acosh(c + d*x)** 
4/(c**2 + 2*c*d*x + d**2*x**2),x)*b**4*c*d*x + 4*int(acosh(c + d*x)**3/(c* 
*2 + 2*c*d*x + d**2*x**2),x)*a*b**3*c**2 + 4*int(acosh(c + d*x)**3/(c**2 + 
 2*c*d*x + d**2*x**2),x)*a*b**3*c*d*x + 6*int(acosh(c + d*x)**2/(c**2 + 2* 
c*d*x + d**2*x**2),x)*a**2*b**2*c**2 + 6*int(acosh(c + d*x)**2/(c**2 + 2*c 
*d*x + d**2*x**2),x)*a**2*b**2*c*d*x + a**4*x)/(c*e**2*(c + d*x))