\(\int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx\) [168]

Optimal result

Integrand size = 20, antiderivative size = 436 \[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^3}+\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^4}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^4} \] Output:

(d*x+c)^3*ln(1+b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f-(d*x+c) 
^3*ln(1+b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f+3*d*(d*x+c)^2* 
polylog(2,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f^2-3*d*(d*x+ 
c)^2*polylog(2,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f^2-6*d^ 
2*(d*x+c)*polylog(3,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f^3 
+6*d^2*(d*x+c)*polylog(3,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2 
)/f^3+6*d^3*polylog(4,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f 
^4-6*d^3*polylog(4,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(1/2)/f^4
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.77 \[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )-(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )+\frac {3 d \left (f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2-b^2}}\right )-2 d f (c+d x) \operatorname {PolyLog}\left (3,\frac {b e^{e+f x}}{-a+\sqrt {a^2-b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (4,\frac {b e^{e+f x}}{-a+\sqrt {a^2-b^2}}\right )\right )}{f^3}-\frac {3 d \left (f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )-2 d f (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )\right )}{f^3}}{\sqrt {a^2-b^2} f} \] Input:

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

Output:

((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])] - (c + d*x)^3* 
Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])] + (3*d*(f^2*(c + d*x)^2*Pol 
yLog[2, (b*E^(e + f*x))/(-a + Sqrt[a^2 - b^2])] - 2*d*f*(c + d*x)*PolyLog[ 
3, (b*E^(e + f*x))/(-a + Sqrt[a^2 - b^2])] + 2*d^2*PolyLog[4, (b*E^(e + f* 
x))/(-a + Sqrt[a^2 - b^2])]))/f^3 - (3*d*(f^2*(c + d*x)^2*PolyLog[2, -((b* 
E^(e + f*x))/(a + Sqrt[a^2 - b^2]))] - 2*d*f*(c + d*x)*PolyLog[3, -((b*E^( 
e + f*x))/(a + Sqrt[a^2 - b^2]))] + 2*d^2*PolyLog[4, -((b*E^(e + f*x))/(a 
+ Sqrt[a^2 - b^2]))]))/f^3)/(Sqrt[a^2 - b^2]*f)
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 3801, 2694, 27, 2620, 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.

\(\displaystyle \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^3}{a+b \sin \left (i e+i f x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3801

\(\displaystyle 2 \int \frac {e^{e+f x} (c+d x)^3}{2 e^{e+f x} a+b e^{2 (e+f x)}+b}dx\)

\(\Big \downarrow \) 2694

\(\displaystyle 2 \left (\frac {b \int \frac {e^{e+f x} (c+d x)^3}{2 \left (a+b e^{e+f x}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {b \int \frac {e^{e+f x} (c+d x)^3}{2 \left (a+b e^{e+f x}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {b \int \frac {e^{e+f x} (c+d x)^3}{a+b e^{e+f x}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {e^{e+f x} (c+d x)^3}{a+b e^{e+f x}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \int (c+d x)^2 \log \left (\frac {e^{e+f x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \int (c+d x)^2 \log \left (\frac {e^{e+f x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle 2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle 2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )de^{e+f x}}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )de^{e+f x}}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 2 \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )\)

Input:

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

Output:

2*((b*(((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/(b*f) 
- (3*d*(-(((c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])) 
])/f) + (2*d*(((c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2] 
))])/f - (d*PolyLog[4, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/f^2))/f) 
)/(b*f)))/(2*Sqrt[a^2 - b^2]) - (b*(((c + d*x)^3*Log[1 + (b*E^(e + f*x))/( 
a + Sqrt[a^2 - b^2])])/(b*f) - (3*d*(-(((c + d*x)^2*PolyLog[2, -((b*E^(e + 
 f*x))/(a + Sqrt[a^2 - b^2]))])/f) + (2*d*(((c + d*x)*PolyLog[3, -((b*E^(e 
 + f*x))/(a + Sqrt[a^2 - b^2]))])/f - (d*PolyLog[4, -((b*E^(e + f*x))/(a + 
 Sqrt[a^2 - b^2]))])/f^2))/f))/(b*f)))/(2*Sqrt[a^2 - b^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 2620

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] - Si 
mp[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 2694

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]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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 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 3801

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple 
x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[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 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]

\[\int \frac {\left (d x +c \right )^{3}}{a +b \cosh \left (f x +e \right )}d x\]

Input:

int((d*x+c)^3/(a+b*cosh(f*x+e)),x)
 

Output:

int((d*x+c)^3/(a+b*cosh(f*x+e)),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (394) = 788\).

Time = 0.13 (sec) , antiderivative size = 1042, normalized size of antiderivative = 2.39 \[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3/(a+b*cosh(f*x+e)),x, algorithm="fricas")
 

Output:

(6*b*d^3*sqrt((a^2 - b^2)/b^2)*polylog(4, -(a*cosh(f*x + e) + a*sinh(f*x + 
 e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 - b^2)/b^2))/b) - 6*b* 
d^3*sqrt((a^2 - b^2)/b^2)*polylog(4, -(a*cosh(f*x + e) + a*sinh(f*x + e) - 
 (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 - b^2)/b^2))/b) + 3*(b*d^3* 
f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*sqrt((a^2 - b^2)/b^2)*dilog(-(a*c 
osh(f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt( 
(a^2 - b^2)/b^2) + b)/b + 1) - 3*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2* 
d*f^2)*sqrt((a^2 - b^2)/b^2)*dilog(-(a*cosh(f*x + e) + a*sinh(f*x + e) - ( 
b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + (b* 
d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sqrt((a^2 - b^2)/ 
b^2)*log(2*b*cosh(f*x + e) + 2*b*sinh(f*x + e) + 2*b*sqrt((a^2 - b^2)/b^2) 
 + 2*a) - (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sqrt 
((a^2 - b^2)/b^2)*log(2*b*cosh(f*x + e) + 2*b*sinh(f*x + e) - 2*b*sqrt((a^ 
2 - b^2)/b^2) + 2*a) + (b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3* 
x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*sqrt((a^2 - b^2)/b^2)*l 
og((a*cosh(f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e) 
)*sqrt((a^2 - b^2)/b^2) + b)/b) - (b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b 
*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*sqrt((a^2 - 
b^2)/b^2)*log((a*cosh(f*x + e) + a*sinh(f*x + e) - (b*cosh(f*x + e) + b*si 
nh(f*x + e))*sqrt((a^2 - b^2)/b^2) + b)/b) - 6*(b*d^3*f*x + b*c*d^2*f)*...
 

Sympy [F]

\[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\int \frac {\left (c + d x\right )^{3}}{a + b \cosh {\left (e + f x \right )}}\, dx \] Input:

integrate((d*x+c)**3/(a+b*cosh(f*x+e)),x)
 

Output:

Integral((c + d*x)**3/(a + b*cosh(e + f*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^3/(a+b*cosh(f*x+e)),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(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [F]

\[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{b \cosh \left (f x + e\right ) + a} \,d x } \] Input:

integrate((d*x+c)^3/(a+b*cosh(f*x+e)),x, algorithm="giac")
 

Output:

integrate((d*x + c)^3/(b*cosh(f*x + e) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (e+f\,x\right )} \,d x \] Input:

int((c + d*x)^3/(a + b*cosh(e + f*x)),x)
 

Output:

int((c + d*x)^3/(a + b*cosh(e + f*x)), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^3}{a+b \cosh (e+f x)} \, dx=\frac {-2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{f x +e} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) c^{3}+2 e^{e} \left (\int \frac {e^{f x} x^{3}}{e^{2 f x +2 e} b +2 e^{f x +e} a +b}d x \right ) a^{2} d^{3} f -2 e^{e} \left (\int \frac {e^{f x} x^{3}}{e^{2 f x +2 e} b +2 e^{f x +e} a +b}d x \right ) b^{2} d^{3} f +6 e^{e} \left (\int \frac {e^{f x} x^{2}}{e^{2 f x +2 e} b +2 e^{f x +e} a +b}d x \right ) a^{2} c \,d^{2} f -6 e^{e} \left (\int \frac {e^{f x} x^{2}}{e^{2 f x +2 e} b +2 e^{f x +e} a +b}d x \right ) b^{2} c \,d^{2} f +6 e^{e} \left (\int \frac {e^{f x} x}{e^{2 f x +2 e} b +2 e^{f x +e} a +b}d x \right ) a^{2} c^{2} d f -6 e^{e} \left (\int \frac {e^{f x} x}{e^{2 f x +2 e} b +2 e^{f x +e} a +b}d x \right ) b^{2} c^{2} d f}{f \left (a^{2}-b^{2}\right )} \] Input:

int((d*x+c)^3/(a+b*cosh(f*x+e)),x)
 

Output:

(2*( - sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2) 
)*c**3 + e**e*int((e**(f*x)*x**3)/(e**(2*e + 2*f*x)*b + 2*e**(e + f*x)*a + 
 b),x)*a**2*d**3*f - e**e*int((e**(f*x)*x**3)/(e**(2*e + 2*f*x)*b + 2*e**( 
e + f*x)*a + b),x)*b**2*d**3*f + 3*e**e*int((e**(f*x)*x**2)/(e**(2*e + 2*f 
*x)*b + 2*e**(e + f*x)*a + b),x)*a**2*c*d**2*f - 3*e**e*int((e**(f*x)*x**2 
)/(e**(2*e + 2*f*x)*b + 2*e**(e + f*x)*a + b),x)*b**2*c*d**2*f + 3*e**e*in 
t((e**(f*x)*x)/(e**(2*e + 2*f*x)*b + 2*e**(e + f*x)*a + b),x)*a**2*c**2*d* 
f - 3*e**e*int((e**(f*x)*x)/(e**(2*e + 2*f*x)*b + 2*e**(e + f*x)*a + b),x) 
*b**2*c**2*d*f))/(f*(a**2 - b**2))