3.1.74 \(\int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx\) [74]

3.1.74.1 Optimal result

Integrand size = 20, antiderivative size = 476 \[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=-\frac {2 b^2 (c+d x)^2}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^3}{3 (a-b)^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {2 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {2 b^2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {b d^2 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {b^2 d^2 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3} \]

output

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

3.1.74.2 Mathematica [A] (verified)

Time = 3.55 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=\frac {24 b c f^2 (-b d+a c f) x-\frac {24 (a-b) b c f^2 (-b d+a c f) x}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )}-\frac {12 (a-b) b d f^2 (-b d+2 a c f) x^2}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )}-\frac {8 a (a-b) b d^2 f^3 x^3}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )}+12 b d f (b d-2 a c f) x \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )-12 a b d^2 f^2 x^2 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+12 b c f (b d-a c f) \log \left (a-b+(a+b) e^{2 (e+f x)}\right )-6 b d (b d-2 a c f) \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+6 a b d^2 \left (2 f x \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+\frac {(a-b) (a+b) f^2 \left (\left (a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (f x)+\left (a^2-b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (2 e+f x)+2 b \left (-3 b (c+d x)^2+a f x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \sinh (f x)\right )}{(a \cosh (e)+b \sinh (e)) (a \cosh (e+f x)+b \sinh (e+f x))}}{6 (a-b)^2 (a+b)^2 f^3} \]

input

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

output

(24*b*c*f^2*(-(b*d) + a*c*f)*x - (24*(a - b)*b*c*f^2*(-(b*d) + a*c*f)*x)/( 
b*(-1 + E^(2*e)) + a*(1 + E^(2*e))) - (12*(a - b)*b*d*f^2*(-(b*d) + 2*a*c* 
f)*x^2)/(b*(-1 + E^(2*e)) + a*(1 + E^(2*e))) - (8*a*(a - b)*b*d^2*f^3*x^3) 
/(b*(-1 + E^(2*e)) + a*(1 + E^(2*e))) + 12*b*d*f*(b*d - 2*a*c*f)*x*Log[1 + 
 (a - b)/((a + b)*E^(2*(e + f*x)))] - 12*a*b*d^2*f^2*x^2*Log[1 + (a - b)/( 
(a + b)*E^(2*(e + f*x)))] + 12*b*c*f*(b*d - a*c*f)*Log[a - b + (a + b)*E^( 
2*(e + f*x))] - 6*b*d*(b*d - 2*a*c*f)*PolyLog[2, (-a + b)/((a + b)*E^(2*(e 
 + f*x)))] + 6*a*b*d^2*(2*f*x*PolyLog[2, (-a + b)/((a + b)*E^(2*(e + f*x)) 
)] + PolyLog[3, (-a + b)/((a + b)*E^(2*(e + f*x)))]) + ((a - b)*(a + b)*f^ 
2*((a^2 + b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cosh[f*x] + (a^2 - b^2)*f*x 
*(3*c^2 + 3*c*d*x + d^2*x^2)*Cosh[2*e + f*x] + 2*b*(-3*b*(c + d*x)^2 + a*f 
*x*(3*c^2 + 3*c*d*x + d^2*x^2))*Sinh[f*x]))/((a*Cosh[e] + b*Sinh[e])*(a*Co 
sh[e + f*x] + b*Sinh[e + f*x])))/(6*(a - b)^2*(a + b)^2*f^3)
 

3.1.74.3 Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4217, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4217

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.1.74.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4217

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + 
 b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
 

3.1.74.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1604\) vs. \(2(465)=930\).

Time = 0.34 (sec) , antiderivative size = 1605, normalized size of antiderivative = 3.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(1605\)

input

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

output

d/(a^2+2*a*b+b^2)*c*x^2+1/(a^2+2*a*b+b^2)*c^2*x+1/3*d^2/(a^2+2*a*b+b^2)*x^ 
3+1/3/d/(a^2+2*a*b+b^2)*c^3-8/(a^2+2*a*b+b^2)/f*b/(a-b)*c*d*a/(-a+b)*e*x+4 
/(a^2+2*a*b+b^2)/f^2*b/(a-b)*c*d*a/(-a+b)*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b) 
)*e+4/(a^2+2*a*b+b^2)/f*b/(a-b)*c*d*a/(-a+b)*ln(1-(a+b)*exp(2*f*x+2*e)/(-a 
+b))*x+2/(a^2+2*a*b+b^2)/f*b^2/(a-b)/(-a+b)*d^2*x^2+2/(a^2+2*a*b+b^2)/f^3* 
b^2/(a-b)/(-a+b)*d^2*e^2+4/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*c^2*ln(exp(f*x+e) 
)-2/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*c^2*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e) 
+a-b)-4/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)^2*c*d*ln(exp(f*x+e))+2/(a^2+2*a*b+b^ 
2)/f^2*b^2/(a-b)^2*c*d*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)-1/(a^2+2* 
a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^2*polylog(2,(a+b)*exp(2*f*x+2*e)/(-a+b))+4 
/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)^2*e*d^2*ln(exp(f*x+e))-2/(a^2+2*a*b+b^2)/f^ 
3*b^2/(a-b)^2*e*d^2*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)-4/3/(a^2+2*a 
*b+b^2)*b/(a-b)/(-a+b)*a*d^2*x^3+8/3/(a^2+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*a* 
d^2*e^3+4/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)/(-a+b)*d^2*e*x-1/(a^2+2*a*b+b^2)/f 
^3*b/(a-b)/(-a+b)*a*d^2*polylog(3,(a+b)*exp(2*f*x+2*e)/(-a+b))-2/(a^2+2*a* 
b+b^2)/f^2*b^2/(a-b)/(-a+b)*d^2*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x-2/(a^2 
+2*a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^2*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*e+4 
/(a^2+2*a*b+b^2)/f^3*b/(a-b)^2*e^2*a*d^2*ln(exp(f*x+e))-2/(a^2+2*a*b+b^2)/ 
f^3*b/(a-b)^2*e^2*a*d^2*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)+2/(a-b)/ 
f/(a^2+2*a*b+b^2)*(d^2*x^2+2*c*d*x+c^2)*b^2/(exp(2*f*x+2*e)*a+b*exp(2*f...
 

3.1.74.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3693 vs. \(2 (462) = 924\).

Time = 0.35 (sec) , antiderivative size = 3693, normalized size of antiderivative = 7.76 \[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \]

input

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

output

1/3*((a^3 + a^2*b - a*b^2 - b^3)*d^2*f^3*x^3 + 3*(a^3 + a^2*b - a*b^2 - b^ 
3)*c*d*f^3*x^2 + 3*(a^3 + a^2*b - a*b^2 - b^3)*c^2*f^3*x + 4*(a^2*b - a*b^ 
2)*d^2*e^3 + 6*(a*b^2 - b^3)*d^2*e^2 + 6*(2*(a^2*b - a*b^2)*c^2*e + (a*b^2 
 - b^3)*c^2)*f^2 + ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^2*f^3*x^3 + 4*(a^2*b 
 + a*b^2)*d^2*e^3 + 12*(a^2*b + a*b^2)*c^2*e*f^2 + 6*(a*b^2 + b^3)*d^2*e^2 
 + 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*d*f^3 - 2*(a*b^2 + b^3)*d^2*f^2)*x 
^2 - 12*((a^2*b + a*b^2)*c*d*e^2 + (a*b^2 + b^3)*c*d*e)*f + 3*((a^3 + 3*a^ 
2*b + 3*a*b^2 + b^3)*c^2*f^3 - 4*(a*b^2 + b^3)*c*d*f^2)*x)*cosh(f*x + e)^2 
 + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^2*f^3*x^3 + 4*(a^2*b + a*b^2)*d^2* 
e^3 + 12*(a^2*b + a*b^2)*c^2*e*f^2 + 6*(a*b^2 + b^3)*d^2*e^2 + 3*((a^3 + 3 
*a^2*b + 3*a*b^2 + b^3)*c*d*f^3 - 2*(a*b^2 + b^3)*d^2*f^2)*x^2 - 12*((a^2* 
b + a*b^2)*c*d*e^2 + (a*b^2 + b^3)*c*d*e)*f + 3*((a^3 + 3*a^2*b + 3*a*b^2 
+ b^3)*c^2*f^3 - 4*(a*b^2 + b^3)*c*d*f^2)*x)*cosh(f*x + e)*sinh(f*x + e) + 
 ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^2*f^3*x^3 + 4*(a^2*b + a*b^2)*d^2*e^3 
+ 12*(a^2*b + a*b^2)*c^2*e*f^2 + 6*(a*b^2 + b^3)*d^2*e^2 + 3*((a^3 + 3*a^2 
*b + 3*a*b^2 + b^3)*c*d*f^3 - 2*(a*b^2 + b^3)*d^2*f^2)*x^2 - 12*((a^2*b + 
a*b^2)*c*d*e^2 + (a*b^2 + b^3)*c*d*e)*f + 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^ 
3)*c^2*f^3 - 4*(a*b^2 + b^3)*c*d*f^2)*x)*sinh(f*x + e)^2 - 12*((a^2*b - a* 
b^2)*c*d*e^2 + (a*b^2 - b^3)*c*d*e)*f - 6*(2*(a^2*b - a*b^2)*d^2*f*x + 2*( 
a^2*b - a*b^2)*c*d*f - (a*b^2 - b^3)*d^2 + (2*(a^2*b + a*b^2)*d^2*f*x +...
 

3.1.74.6 Sympy [F]

\[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (a + b \tanh {\left (e + f x \right )}\right )^{2}}\, dx \]

input

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

output

Integral((c + d*x)**2/(a + b*tanh(e + f*x))**2, x)
 

3.1.74.7 Maxima [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=-\frac {4 \, b^{2} c d f x}{a^{4} f^{2} - 2 \, a^{2} b^{2} f^{2} + b^{4} f^{2}} - \frac {{\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 )} a b d^{2}}{a^{4} f^{3} - 2 \, a^{2} b^{2} f^{3} + b^{4} f^{3}} + \frac {2 \, b^{2} c d \log \left ({\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + a - b\right )}{a^{4} f^{2} - 2 \, a^{2} b^{2} f^{2} + b^{4} f^{2}} - c^{2} {\left (\frac {2 \, a b \log \left (-{\left (a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} - a - b\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} f} + \frac {2 \, b^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )}\right )} f} - \frac {f x + e}{{\left (a^{2} + 2 \, a b + b^{2}\right )} f}\right )} - \frac {{\left (2 \, a b c d f - b^{2} d^{2}\right )} {\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 )}}{a^{4} f^{3} - 2 \, a^{2} b^{2} f^{3} + b^{4} f^{3}} + \frac {2 \, {\left (2 \, a b d^{2} f^{3} x^{3} + 3 \, {\left (2 \, a b c d f - b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, {\left (a^{4} f^{3} - 2 \, a^{2} b^{2} f^{3} + b^{4} f^{3}\right )}} + \frac {12 \, b^{2} c d x + {\left (a^{2} d^{2} f - 2 \, a b d^{2} f + b^{2} d^{2} f\right )} x^{3} + 3 \, {\left (a^{2} c d f - 2 \, a b c d f + {\left (c d f + 2 \, d^{2}\right )} b^{2}\right )} x^{2} + {\left ({\left (a^{2} d^{2} f e^{\left (2 \, e\right )} - b^{2} d^{2} f e^{\left (2 \, e\right )}\right )} x^{3} + 3 \, {\left (a^{2} c d f e^{\left (2 \, e\right )} - b^{2} c d f e^{\left (2 \, e\right )}\right )} x^{2}\right )} e^{\left (2 \, f x\right )}}{3 \, {\left (a^{4} f - 2 \, a^{2} b^{2} f + b^{4} f + {\left (a^{4} f e^{\left (2 \, e\right )} + 2 \, a^{3} b f e^{\left (2 \, e\right )} - 2 \, a b^{3} f e^{\left (2 \, e\right )} - b^{4} f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}\right )}} \]

input

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

output

-4*b^2*c*d*f*x/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^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*dilog(-(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)))*a*b*d^2/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + 2*b^2*c*d*log((a*e^ 
(2*e) + b*e^(2*e))*e^(2*f*x) + a - b)/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^2) 
- c^2*(2*a*b*log(-(a - b)*e^(-2*f*x - 2*e) - a - b)/((a^4 - 2*a^2*b^2 + b^ 
4)*f) + 2*b^2/((a^4 - 2*a^2*b^2 + b^4 + (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^ 
(-2*f*x - 2*e))*f) - (f*x + e)/((a^2 + 2*a*b + b^2)*f)) - (2*a*b*c*d*f - b 
^2*d^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)))/(a^4*f^3 - 2*a^2*b^2*f^3 + b^ 
4*f^3) + 2/3*(2*a*b*d^2*f^3*x^3 + 3*(2*a*b*c*d*f - b^2*d^2)*f^2*x^2)/(a^4* 
f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + 1/3*(12*b^2*c*d*x + (a^2*d^2*f - 2*a*b*d^ 
2*f + b^2*d^2*f)*x^3 + 3*(a^2*c*d*f - 2*a*b*c*d*f + (c*d*f + 2*d^2)*b^2)*x 
^2 + ((a^2*d^2*f*e^(2*e) - b^2*d^2*f*e^(2*e))*x^3 + 3*(a^2*c*d*f*e^(2*e) - 
 b^2*c*d*f*e^(2*e))*x^2)*e^(2*f*x))/(a^4*f - 2*a^2*b^2*f + b^4*f + (a^4*f* 
e^(2*e) + 2*a^3*b*f*e^(2*e) - 2*a*b^3*f*e^(2*e) - b^4*f*e^(2*e))*e^(2*f*x) 
)
 

3.1.74.8 Giac [F]

\[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

input

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

output

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

3.1.74.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2} \,d x \]

input

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

output

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