\(\int \frac {(c+d x)^3}{a+b \tanh (e+f x)} \, dx\) [68]

Optimal result

Integrand size = 20, antiderivative size = 212 \[ \int \frac {(c+d x)^3}{a+b \tanh (e+f x)} \, dx=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{4 \left (a^2-b^2\right ) f^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x)^3}{a+b \tanh (e+f x)} \, dx=\frac {1}{4} \left (-\frac {2 b (c+d x)^4}{(a+b) d \left (b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )\right )}-\frac {4 b (c+d x)^3 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{(a-b) (a+b) f}+\frac {3 b d \left (2 f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+d \left (2 f (c+d x) \operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+d \operatorname {PolyLog}\left (4,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )\right )}{(a-b) (a+b) f^4}+\frac {x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (e)}{a \cosh (e)+b \sinh (e)}\right ) \] Input:

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

Output:

((-2*b*(c + d*x)^4)/((a + b)*d*(b*(-1 + E^(2*e)) + a*(1 + E^(2*e)))) - (4* 
b*(c + d*x)^3*Log[1 + (a - b)/((a + b)*E^(2*(e + f*x)))])/((a - b)*(a + b) 
*f) + (3*b*d*(2*f^2*(c + d*x)^2*PolyLog[2, (-a + b)/((a + b)*E^(2*(e + f*x 
)))] + d*(2*f*(c + d*x)*PolyLog[3, (-a + b)/((a + b)*E^(2*(e + f*x)))] + d 
*PolyLog[4, (-a + b)/((a + b)*E^(2*(e + f*x)))])))/((a - b)*(a + b)*f^4) + 
 (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[e])/(a*Cosh[e] + b*Si 
nh[e]))/4
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 4215, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4215

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 4215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)), x] + Simp[2*I*b   In 
t[(c + d*x)^m*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Simp[2 
*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1155\) vs. \(2(212)=424\).

Time = 0.89 (sec) , antiderivative size = 1156, normalized size of antiderivative = 5.45

method	result	size
risch	\(\text {Expression too large to display}\)	\(1156\)

Input:

int((d*x+c)^3/(a+b*tanh(f*x+e)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (203) = 406\).

Time = 0.11 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.49 \[ \int \frac {(c+d x)^3}{a+b \tanh (e+f x)} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*((a + b)*d^3*f^4*x^4 + 4*(a + b)*c*d^2*f^4*x^3 + 6*(a + b)*c^2*d*f^4*x 
^2 + 4*(a + b)*c^3*f^4*x - 24*b*d^3*polylog(4, sqrt(-(a + b)/(a - b))*(cos 
h(f*x + e) + sinh(f*x + e))) - 24*b*d^3*polylog(4, -sqrt(-(a + b)/(a - b)) 
*(cosh(f*x + e) + sinh(f*x + e))) - 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + 
b*c^2*d*f^2)*dilog(sqrt(-(a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) 
 - 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*dilog(-sqrt(-(a + b) 
/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 4*(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)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*s 
inh(f*x + e) + 2*(a - b)*sqrt(-(a + b)/(a - b))) + 4*(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)*log(2*(a + b)*cosh(f*x + e) + 2*(a 
+ b)*sinh(f*x + e) - 2*(a - b)*sqrt(-(a + b)/(a - b))) - 4*(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)*log(sqrt(-(a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e)) + 
1) - 4*(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)*log(-sqrt(-(a + b)/(a - b))*(cosh(f*x + 
 e) + sinh(f*x + e)) + 1) + 24*(b*d^3*f*x + b*c*d^2*f)*polylog(3, sqrt(-(a 
 + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 24*(b*d^3*f*x + b*c*d^2* 
f)*polylog(3, -sqrt(-(a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))))/(( 
a^2 - b^2)*f^4)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (203) = 406\).

Time = 0.20 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.49 \[ \int \frac {(c+d x)^3}{a+b \tanh (e+f x)} \, dx=-\frac {3 \, {\left (2 \, f x \log \left (\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right )\right )} b c^{2} d}{2 \, {\left (a^{2} f^{2} - b^{2} f^{2}\right )}} - \frac {3 \, {\left (2 \, f^{2} x^{2} \log \left (\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + 2 \, f x {\rm Li}_2\left (-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right ) - {\rm Li}_{3}(-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b})\right )} b c d^{2}}{2 \, {\left (a^{2} f^{3} - b^{2} f^{3}\right )}} - \frac {{\left (4 \, f^{3} x^{3} \log \left (\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right ) - 6 \, f x {\rm Li}_{3}(-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}) + 3 \, {\rm Li}_{4}(-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b})\right )} b d^{3}}{3 \, {\left (a^{2} f^{4} - b^{2} f^{4}\right )}} - c^{3} {\left (\frac {b \log \left (-{\left (a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} - a - b\right )}{{\left (a^{2} - b^{2}\right )} f} - \frac {f x + e}{{\left (a + b\right )} f}\right )} + \frac {b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b c^{2} d f^{4} x^{2}}{2 \, {\left (a^{2} f^{4} - b^{2} f^{4}\right )}} + \frac {d^{3} x^{4} + 4 \, c d^{2} x^{3} + 6 \, c^{2} d x^{2}}{4 \, {\left (a + b\right )}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^3}{a+b \tanh (e+f x)} \, dx=\frac {-8 e^{2 e} \left (\int \frac {e^{2 f x} x^{3}}{e^{2 f x +2 e} a^{2}-e^{2 f x +2 e} b^{2}+a^{2}-2 a b +b^{2}}d x \right ) a^{2} b \,d^{3} f +8 e^{2 e} \left (\int \frac {e^{2 f x} x^{3}}{e^{2 f x +2 e} a^{2}-e^{2 f x +2 e} b^{2}+a^{2}-2 a b +b^{2}}d x \right ) b^{3} d^{3} f -24 e^{2 e} \left (\int \frac {e^{2 f x} x^{2}}{e^{2 f x +2 e} a^{2}-e^{2 f x +2 e} b^{2}+a^{2}-2 a b +b^{2}}d x \right ) a^{2} b c \,d^{2} f +24 e^{2 e} \left (\int \frac {e^{2 f x} x^{2}}{e^{2 f x +2 e} a^{2}-e^{2 f x +2 e} b^{2}+a^{2}-2 a b +b^{2}}d x \right ) b^{3} c \,d^{2} f -24 e^{2 e} \left (\int \frac {e^{2 f x} x}{e^{2 f x +2 e} a^{2}-e^{2 f x +2 e} b^{2}+a^{2}-2 a b +b^{2}}d x \right ) a^{2} b \,c^{2} d f +24 e^{2 e} \left (\int \frac {e^{2 f x} x}{e^{2 f x +2 e} a^{2}-e^{2 f x +2 e} b^{2}+a^{2}-2 a b +b^{2}}d x \right ) b^{3} c^{2} d f -4 \,\mathrm {log}\left (e^{2 f x +2 e} a +e^{2 f x +2 e} b +a -b \right ) b \,c^{3}+4 a \,c^{3} f x +6 a \,c^{2} d f \,x^{2}+4 a c \,d^{2} f \,x^{3}+a \,d^{3} f \,x^{4}+4 b \,c^{3} f x +6 b \,c^{2} d f \,x^{2}+4 b c \,d^{2} f \,x^{3}+b \,d^{3} f \,x^{4}}{4 f \left (a^{2}-b^{2}\right )} \] Input:

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

Output:

( - 8*e**(2*e)*int((e**(2*f*x)*x**3)/(e**(2*e + 2*f*x)*a**2 - e**(2*e + 2* 
f*x)*b**2 + a**2 - 2*a*b + b**2),x)*a**2*b*d**3*f + 8*e**(2*e)*int((e**(2* 
f*x)*x**3)/(e**(2*e + 2*f*x)*a**2 - e**(2*e + 2*f*x)*b**2 + a**2 - 2*a*b + 
 b**2),x)*b**3*d**3*f - 24*e**(2*e)*int((e**(2*f*x)*x**2)/(e**(2*e + 2*f*x 
)*a**2 - e**(2*e + 2*f*x)*b**2 + a**2 - 2*a*b + b**2),x)*a**2*b*c*d**2*f + 
 24*e**(2*e)*int((e**(2*f*x)*x**2)/(e**(2*e + 2*f*x)*a**2 - e**(2*e + 2*f* 
x)*b**2 + a**2 - 2*a*b + b**2),x)*b**3*c*d**2*f - 24*e**(2*e)*int((e**(2*f 
*x)*x)/(e**(2*e + 2*f*x)*a**2 - e**(2*e + 2*f*x)*b**2 + a**2 - 2*a*b + b** 
2),x)*a**2*b*c**2*d*f + 24*e**(2*e)*int((e**(2*f*x)*x)/(e**(2*e + 2*f*x)*a 
**2 - e**(2*e + 2*f*x)*b**2 + a**2 - 2*a*b + b**2),x)*b**3*c**2*d*f - 4*lo 
g(e**(2*e + 2*f*x)*a + e**(2*e + 2*f*x)*b + a - b)*b*c**3 + 4*a*c**3*f*x + 
 6*a*c**2*d*f*x**2 + 4*a*c*d**2*f*x**3 + a*d**3*f*x**4 + 4*b*c**3*f*x + 6* 
b*c**2*d*f*x**2 + 4*b*c*d**2*f*x**3 + b*d**3*f*x**4)/(4*f*(a**2 - b**2))