\(\int \frac {x^3}{a+b \cosh ^2(x)} \, dx\) [212]

Optimal result

Integrand size = 14, antiderivative size = 391 \[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}-\frac {x^3 \log \left (1+\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{2 \sqrt {a} \sqrt {a+b}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {3 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 x \operatorname {PolyLog}\left (3,-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{4 \sqrt {a} \sqrt {a+b}}+\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+b-2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}}-\frac {3 \operatorname {PolyLog}\left (4,-\frac {b e^{2 x}}{2 a+b+2 \sqrt {a} \sqrt {a+b}}\right )}{8 \sqrt {a} \sqrt {a+b}} \] Output:

1/2*x^3*ln(1+b*exp(2*x)/(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2) 
-1/2*x^3*ln(1+b*exp(2*x)/(2*a+b+2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+b)^(1/2 
)+3/4*x^2*polylog(2,-b*exp(2*x)/(2*a+b-2*a^(1/2)*(a+b)^(1/2)))/a^(1/2)/(a+ 
b)^(1/2)-3/4*x^2*polylog(2,-b*exp(2*x)/(2*a+b+2*a^(1/2)*(a+b)^(1/2)))/a^(1 
/2)/(a+b)^(1/2)-3/4*x*polylog(3,-b*exp(2*x)/(2*a+b-2*a^(1/2)*(a+b)^(1/2))) 
/a^(1/2)/(a+b)^(1/2)+3/4*x*polylog(3,-b*exp(2*x)/(2*a+b+2*a^(1/2)*(a+b)^(1 
/2)))/a^(1/2)/(a+b)^(1/2)+3/8*polylog(4,-b*exp(2*x)/(2*a+b-2*a^(1/2)*(a+b) 
^(1/2)))/a^(1/2)/(a+b)^(1/2)-3/8*polylog(4,-b*exp(2*x)/(2*a+b+2*a^(1/2)*(a 
+b)^(1/2)))/a^(1/2)/(a+b)^(1/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.48 \[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\frac {x^3 \log \left (1-\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )+x^3 \log \left (1+\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )-x^3 \log \left (1-\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )-x^3 \log \left (1+\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )+3 x^2 \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )+3 x^2 \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )-3 x^2 \operatorname {PolyLog}\left (2,-\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )-3 x^2 \operatorname {PolyLog}\left (2,\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )-6 x \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )-6 x \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )+6 x \operatorname {PolyLog}\left (3,-\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )+6 x \operatorname {PolyLog}\left (3,\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )+6 \operatorname {PolyLog}\left (4,-\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )+6 \operatorname {PolyLog}\left (4,\frac {e^x}{\sqrt {-\frac {2 a+b-2 \sqrt {a (a+b)}}{b}}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )-6 \operatorname {PolyLog}\left (4,\frac {e^x}{\sqrt {-\frac {2 a+b+2 \sqrt {a (a+b)}}{b}}}\right )}{2 \sqrt {a (a+b)}} \] Input:

Integrate[x^3/(a + b*Cosh[x]^2),x]
 

Output:

(x^3*Log[1 - E^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a + b)])/b)]] + x^3*Log[1 + E 
^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a + b)])/b)]] - x^3*Log[1 - E^x/Sqrt[-((2*a 
 + b + 2*Sqrt[a*(a + b)])/b)]] - x^3*Log[1 + E^x/Sqrt[-((2*a + b + 2*Sqrt[ 
a*(a + b)])/b)]] + 3*x^2*PolyLog[2, -(E^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a + 
b)])/b)])] + 3*x^2*PolyLog[2, E^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a + b)])/b)] 
] - 3*x^2*PolyLog[2, -(E^x/Sqrt[-((2*a + b + 2*Sqrt[a*(a + b)])/b)])] - 3* 
x^2*PolyLog[2, E^x/Sqrt[-((2*a + b + 2*Sqrt[a*(a + b)])/b)]] - 6*x*PolyLog 
[3, -(E^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a + b)])/b)])] - 6*x*PolyLog[3, E^x/ 
Sqrt[-((2*a + b - 2*Sqrt[a*(a + b)])/b)]] + 6*x*PolyLog[3, -(E^x/Sqrt[-((2 
*a + b + 2*Sqrt[a*(a + b)])/b)])] + 6*x*PolyLog[3, E^x/Sqrt[-((2*a + b + 2 
*Sqrt[a*(a + b)])/b)]] + 6*PolyLog[4, -(E^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a 
+ b)])/b)])] + 6*PolyLog[4, E^x/Sqrt[-((2*a + b - 2*Sqrt[a*(a + b)])/b)]] 
- 6*PolyLog[4, -(E^x/Sqrt[-((2*a + b + 2*Sqrt[a*(a + b)])/b)])] - 6*PolyLo 
g[4, E^x/Sqrt[-((2*a + b + 2*Sqrt[a*(a + b)])/b)]])/(2*Sqrt[a*(a + b)])
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6164, 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.

\(\displaystyle \int \frac {x^3}{a+b \cosh ^2(x)} \, dx\)

\(\Big \downarrow \) 6164

\(\displaystyle 2 \int \frac {x^3}{2 a+b+b \cosh (2 x)}dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3801

\(\displaystyle 4 \int \frac {e^{2 x} x^3}{e^{4 x} b+b+2 (2 a+b) e^{2 x}}dx\)

\(\Big \downarrow \) 2694

\(\displaystyle 4 \left (\frac {b \int \frac {e^{2 x} x^3}{2 \left (2 a-2 \sqrt {a+b} \sqrt {a}+b e^{2 x}+b\right )}dx}{2 \sqrt {a} \sqrt {a+b}}-\frac {b \int \frac {e^{2 x} x^3}{2 \left (2 a+2 \sqrt {a+b} \sqrt {a}+b e^{2 x}+b\right )}dx}{2 \sqrt {a} \sqrt {a+b}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \left (\frac {b \int \frac {e^{2 x} x^3}{2 a-2 \sqrt {a+b} \sqrt {a}+b e^{2 x}+b}dx}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \int \frac {e^{2 x} x^3}{2 a+2 \sqrt {a+b} \sqrt {a}+b e^{2 x}+b}dx}{4 \sqrt {a} \sqrt {a+b}}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle 4 \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{-2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 b}-\frac {3 \int x^2 \log \left (\frac {e^{2 x} b}{2 a-2 \sqrt {a+b} \sqrt {a}+b}+1\right )dx}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{2 x}}{2 \sqrt {a} \sqrt {a+b}+2 a+b}+1\right )}{2 b}-\frac {3 \int x^2 \log \left (\frac {e^{2 x} b}{2 a+2 \sqrt {a+b} \sqrt {a}+b}+1\right )dx}{2 b}\right )}{4 \sqrt {a} \sqrt {a+b}}\right )\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[x^3/(a + b*Cosh[x]^2),x]
 

Output:

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

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 6164

Int[(Cosh[(c_.) + (d_.)*(x_)]^2*(b_.) + (a_))^(n_)*(x_)^(m_.), x_Symbol] :> 
 Simp[1/2^n   Int[x^m*(2*a + b + b*Cosh[2*c + 2*d*x])^n, x], x] /; FreeQ[{a 
, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] 
|| (EqQ[m, 1] && EqQ[n, -2]))
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(888\) vs. \(2(303)=606\).

Time = 0.54 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(889\)

Input:

int(x^3/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*x 
^3-1/2/(-2*(a*(a+b))^(1/2)-2*a-b)*x^4+1/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2 
)-2*a-b)*ln(1-b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*a*x^3+1/2/(a*(a+b))^( 
1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*ln(1-b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b) 
)*b*x^3-1/2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*a*x^4-1/4/(a*(a+b)) 
^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*b*x^4+3/2/(-2*(a*(a+b))^(1/2)-2*a-b)*pol 
ylog(2,b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*x^2+3/2/(a*(a+b))^(1/2)/(-2* 
(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*a* 
x^2+3/4/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(2,b*exp(2*x)/(- 
2*(a*(a+b))^(1/2)-2*a-b))*b*x^2-3/2/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(3,b 
*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*x-3/2/(a*(a+b))^(1/2)/(-2*(a*(a+b))^ 
(1/2)-2*a-b)*polylog(3,b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*a*x-3/4/(a*( 
a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(3,b*exp(2*x)/(-2*(a*(a+b))^ 
(1/2)-2*a-b))*b*x+3/4/(-2*(a*(a+b))^(1/2)-2*a-b)*polylog(4,b*exp(2*x)/(-2* 
(a*(a+b))^(1/2)-2*a-b))+3/4/(a*(a+b))^(1/2)/(-2*(a*(a+b))^(1/2)-2*a-b)*pol 
ylog(4,b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*a+3/8/(a*(a+b))^(1/2)/(-2*(a 
*(a+b))^(1/2)-2*a-b)*polylog(4,b*exp(2*x)/(-2*(a*(a+b))^(1/2)-2*a-b))*b+1/ 
2/(a*(a+b))^(1/2)*x^3*ln(1-b*exp(2*x)/(2*(a*(a+b))^(1/2)-2*a-b))-1/4/(a*(a 
+b))^(1/2)*x^4+3/4/(a*(a+b))^(1/2)*x^2*polylog(2,b*exp(2*x)/(2*(a*(a+b))^( 
1/2)-2*a-b))-3/4/(a*(a+b))^(1/2)*x*polylog(3,b*exp(2*x)/(2*(a*(a+b))^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1542 vs. \(2 (307) = 614\).

Time = 0.13 (sec) , antiderivative size = 1542, normalized size of antiderivative = 3.94 \[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(a+b*cosh(x)^2),x, algorithm="fricas")
 

Output:

-1/2*(b*x^3*sqrt((a^2 + a*b)/b^2)*log((((2*a + b)*cosh(x) + (2*a + b)*sinh 
(x) - 2*(b*cosh(x) + b*sinh(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^ 
2 + a*b)/b^2) + 2*a + b)/b) + b)/b) + b*x^3*sqrt((a^2 + a*b)/b^2)*log(-((( 
2*a + b)*cosh(x) + (2*a + b)*sinh(x) - 2*(b*cosh(x) + b*sinh(x))*sqrt((a^2 
 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b) - b*x 
^3*sqrt((a^2 + a*b)/b^2)*log((((2*a + b)*cosh(x) + (2*a + b)*sinh(x) + 2*( 
b*cosh(x) + b*sinh(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b 
^2) - 2*a - b)/b) + b)/b) - b*x^3*sqrt((a^2 + a*b)/b^2)*log(-(((2*a + b)*c 
osh(x) + (2*a + b)*sinh(x) + 2*(b*cosh(x) + b*sinh(x))*sqrt((a^2 + a*b)/b^ 
2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - b)/b) + 3*b*x^2*sqrt(( 
a^2 + a*b)/b^2)*dilog(-(((2*a + b)*cosh(x) + (2*a + b)*sinh(x) - 2*(b*cosh 
(x) + b*sinh(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 
 2*a + b)/b) + b)/b + 1) + 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog((((2*a + b) 
*cosh(x) + (2*a + b)*sinh(x) - 2*(b*cosh(x) + b*sinh(x))*sqrt((a^2 + a*b)/ 
b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - b)/b + 1) - 3*b*x^2 
*sqrt((a^2 + a*b)/b^2)*dilog(-(((2*a + b)*cosh(x) + (2*a + b)*sinh(x) + 2* 
(b*cosh(x) + b*sinh(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/ 
b^2) - 2*a - b)/b) + b)/b + 1) - 3*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog((((2* 
a + b)*cosh(x) + (2*a + b)*sinh(x) + 2*(b*cosh(x) + b*sinh(x))*sqrt((a^2 + 
 a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - b)/b + 1) -...
 

Sympy [F]

\[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\int \frac {x^{3}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \] Input:

integrate(x**3/(a+b*cosh(x)**2),x)
 

Output:

Integral(x**3/(a + b*cosh(x)**2), x)
 

Maxima [F]

\[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\int { \frac {x^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:

integrate(x^3/(a+b*cosh(x)^2),x, algorithm="maxima")
 

Output:

integrate(x^3/(b*cosh(x)^2 + a), x)
 

Giac [F]

\[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\int { \frac {x^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:

integrate(x^3/(a+b*cosh(x)^2),x, algorithm="giac")
 

Output:

integrate(x^3/(b*cosh(x)^2 + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\int \frac {x^3}{b\,{\mathrm {cosh}\left (x\right )}^2+a} \,d x \] Input:

int(x^3/(a + b*cosh(x)^2),x)
 

Output:

int(x^3/(a + b*cosh(x)^2), x)
 

Reduce [F]

\[ \int \frac {x^3}{a+b \cosh ^2(x)} \, dx=\int \frac {x^{3}}{\cosh \left (x \right )^{2} b +a}d x \] Input:

int(x^3/(a+b*cosh(x)^2),x)
 

Output:

int(x**3/(cosh(x)**2*b + a),x)