\(\int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx\) [180]

Optimal result

Integrand size = 13, antiderivative size = 57 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {\left (a^2-b^2\right ) \log (\cosh (x))}{a^3}-\frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}-\frac {b \text {sech}(x)}{a^2}+\frac {\text {sech}^2(x)}{2 a} \] Output:

(a^2-b^2)*ln(cosh(x))/a^3-(a^2-b^2)*ln(a+b*cosh(x))/a^3-b*sech(x)/a^2+1/2* 
sech(x)^2/a
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {2 \left (a^2-b^2\right ) (\log (\cosh (x))-\log (a+b \cosh (x)))-2 a b \text {sech}(x)+a^2 \text {sech}^2(x)}{2 a^3} \] Input:

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

Output:

(2*(a^2 - b^2)*(Log[Cosh[x]] - Log[a + b*Cosh[x]]) - 2*a*b*Sech[x] + a^2*S 
ech[x]^2)/(2*a^3)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 3200, 522, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {1}{\left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right ) \tan \left (i x-\frac {\pi }{2}\right )^3}dx\)

\(\Big \downarrow \) 3200

\(\displaystyle -\int \frac {\left (b^2-b^2 \cosh ^2(x)\right ) \text {sech}^3(x)}{b^3 (a+b \cosh (x))}d(b \cosh (x))\)

\(\Big \downarrow \) 522

\(\displaystyle -\int \left (\frac {\text {sech}^3(x)}{a b}-\frac {\text {sech}^2(x)}{a^2}+\frac {\left (b^2-a^2\right ) \text {sech}(x)}{a^3 b}+\frac {a^2-b^2}{a^3 (a+b \cosh (x))}\right )d(b \cosh (x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \text {sech}(x)}{a^2}+\frac {\left (a^2-b^2\right ) \log (b \cosh (x))}{a^3}-\frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}+\frac {\text {sech}^2(x)}{2 a}\)

Input:

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

Output:

((a^2 - b^2)*Log[b*Cosh[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cosh[x]])/a^3 - 
(b*Sech[x])/a^2 + Sech[x]^2/(2*a)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

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 3200

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p 
_.), x_Symbol] :> Simp[1/f   Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b 
^2, 0] && IntegerQ[(p + 1)/2]
 

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67

method	result	size
default	\(\frac {\frac {2 a^{2}}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{2}}-\frac {2 a \left (a +b \right )}{\tanh \left (\frac {x}{2}\right )^{2}+1}+\left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )}{a^{3}}-\frac {\left (a +b \right ) \left (a -b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -b \tanh \left (\frac {x}{2}\right )^{2}-a -b \right )}{a^{3}}\)	\(95\)
risch	\(\frac {2 \,{\mathrm e}^{x} \left (-{\mathrm e}^{2 x} b +a \,{\mathrm e}^{x}-b \right )}{\left ({\mathrm e}^{2 x}+1\right )^{2} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+1\right ) b^{2}}{a^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right ) b^{2}}{a^{3}}\)	\(100\)

Input:

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

Output:

1/a^3*(2*a^2/(tanh(1/2*x)^2+1)^2-2*a*(a+b)/(tanh(1/2*x)^2+1)+(a^2-b^2)*ln( 
tanh(1/2*x)^2+1))-(a+b)*(a-b)/a^3*ln(tanh(1/2*x)^2*a-b*tanh(1/2*x)^2-a-b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (55) = 110\).

Time = 0.10 (sec) , antiderivative size = 450, normalized size of antiderivative = 7.89 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=-\frac {2 \, a b \cosh \left (x\right )^{3} + 2 \, a b \sinh \left (x\right )^{3} - 2 \, a^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right ) - a^{2}\right )} \sinh \left (x\right )^{2} + {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right )^{2} - 2 \, a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{a^{3} \cosh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{3} \sinh \left (x\right )^{4} + 2 \, a^{3} \cosh \left (x\right )^{2} + a^{3} + 2 \, {\left (3 \, a^{3} \cosh \left (x\right )^{2} + a^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{3} \cosh \left (x\right )^{3} + a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \] Input:

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

Output:

-(2*a*b*cosh(x)^3 + 2*a*b*sinh(x)^3 - 2*a^2*cosh(x)^2 + 2*a*b*cosh(x) + 2* 
(3*a*b*cosh(x) - a^2)*sinh(x)^2 + ((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*c 
osh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3* 
(a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)* 
cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))*log(2*(b*cosh(x) + a)/(cosh(x) - 
 sinh(x))) - ((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a 
^2 - b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*cosh(x)^2 
 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 + (a^2 - b^ 
2)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) + 2*(3*a*b*cosh(x) 
^2 - 2*a^2*cosh(x) + a*b)*sinh(x))/(a^3*cosh(x)^4 + 4*a^3*cosh(x)*sinh(x)^ 
3 + a^3*sinh(x)^4 + 2*a^3*cosh(x)^2 + a^3 + 2*(3*a^3*cosh(x)^2 + a^3)*sinh 
(x)^2 + 4*(a^3*cosh(x)^3 + a^3*cosh(x))*sinh(x))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=-\frac {2 \, {\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} + b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} + a^{2} e^{\left (-4 \, x\right )} + a^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{3}} \] Input:

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

Output:

-2*(b*e^(-x) - a*e^(-2*x) + b*e^(-3*x))/(2*a^2*e^(-2*x) + a^2*e^(-4*x) + a 
^2) - (a^2 - b^2)*log(2*a*e^(-x) + b*e^(-2*x) + b)/a^3 + (a^2 - b^2)*log(e 
^(-2*x) + 1)/a^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (55) = 110\).

Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.02 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x}\right )}{a^{3}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, a b {\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a^{2}}{2 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \] Input:

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

Output:

(a^2 - b^2)*log(e^(-x) + e^x)/a^3 - (a^2*b - b^3)*log(abs(b*(e^(-x) + e^x) 
 + 2*a))/(a^3*b) - 1/2*(3*a^2*(e^(-x) + e^x)^2 - 3*b^2*(e^(-x) + e^x)^2 + 
4*a*b*(e^(-x) + e^x) - 4*a^2)/(a^3*(e^(-x) + e^x)^2)
 

Mupad [B] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 1221, normalized size of antiderivative = 21.42 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(2/a - (2*b*exp(x))/a^2)/(exp(2*x) + 1) - 2/(a*(2*exp(2*x) + exp(4*x) + 1) 
) + ((2*atan((4*a^4*b^3*(a^2 - b^2)^2*(-a^6)^(1/2) - 4*a^6*b*(a^2 - b^2)^2 
*(-a^6)^(1/2))*(exp(x)*(1/(16*a^4*b^2*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2)) 
 - (a^2 - 2*b^2)^2/(16*a^8*b^2*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2))) + 1/( 
8*a^5*b*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2)) + (a^2 - 2*b^2)/(8*a^7*b*(a^2 
 - b^2)^3*((a^2 - b^2)^2)^(1/2)))) + 2*atan((a^2*(-a^6)^(1/2)*(a^4 + b^4 - 
 2*a^2*b^2)^(1/2) - 2*b^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(2*a 
^3*(a^2 - b^2)^2) + ((a^7 - a^5*b^2)*(-a^6)^(1/2))/(2*a^6*(a^2 - b^2)*((a^ 
2 - b^2)^2)^(1/2)) + (a^6*b^2*exp(3*x)*((2*(a^7 - a^5*b^2)*(a^4 + b^4 - 2* 
a^2*b^2)^(1/2))/(a^11*b^3*(a^2 - b^2)*((a^2 - b^2)^2)^(1/2)) - (2*(a^2 - 2 
*b^2)*(a^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2) - 2*b^2*(-a^6)^(1/2) 
*(a^4 + b^4 - 2*a^2*b^2)^(1/2))*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(a^10*b^3*( 
a^2 - b^2)^2*(-a^6)^(1/2)))*(-a^6)^(1/2))/(8*(a^4 + b^4 - 2*a^2*b^2)^(1/2) 
) - (a^6*b^2*exp(x)*(-a^6)^(1/2)*((8*(a^4 + b^4 - 2*a^2*b^2))/(a^8*b*(a^2 
- b^2)^2) - (4*(2*a^6*b - 2*a^4*b^3)*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(a^12* 
b^2*(a^2 - b^2)*((a^2 - b^2)^2)^(1/2)) - (2*(a^7 - a^5*b^2)*(a^4 + b^4 - 2 
*a^2*b^2)^(1/2))/(a^11*b^3*(a^2 - b^2)*((a^2 - b^2)^2)^(1/2)) + (2*(a^2 - 
2*b^2)*(a^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2) - 2*b^2*(-a^6)^(1/2 
)*(a^4 + b^4 - 2*a^2*b^2)^(1/2))*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(a^10*b^3* 
(a^2 - b^2)^2*(-a^6)^(1/2))))/(8*(a^4 + b^4 - 2*a^2*b^2)^(1/2)) + (a^6*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 292, normalized size of antiderivative = 5.12 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {e^{4 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{2}-e^{4 x} \mathrm {log}\left (e^{2 x}+1\right ) b^{2}-e^{4 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) a^{2}+e^{4 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) b^{2}-e^{4 x} a^{2}-2 e^{3 x} a b +2 e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{2}-2 e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) b^{2}-2 e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) a^{2}+2 e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) b^{2}-2 e^{x} a b +\mathrm {log}\left (e^{2 x}+1\right ) a^{2}-\mathrm {log}\left (e^{2 x}+1\right ) b^{2}-\mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) a^{2}+\mathrm {log}\left (e^{2 x} b +2 e^{x} a +b \right ) b^{2}-a^{2}}{a^{3} \left (e^{4 x}+2 e^{2 x}+1\right )} \] Input:

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

Output:

(e**(4*x)*log(e**(2*x) + 1)*a**2 - e**(4*x)*log(e**(2*x) + 1)*b**2 - e**(4 
*x)*log(e**(2*x)*b + 2*e**x*a + b)*a**2 + e**(4*x)*log(e**(2*x)*b + 2*e**x 
*a + b)*b**2 - e**(4*x)*a**2 - 2*e**(3*x)*a*b + 2*e**(2*x)*log(e**(2*x) + 
1)*a**2 - 2*e**(2*x)*log(e**(2*x) + 1)*b**2 - 2*e**(2*x)*log(e**(2*x)*b + 
2*e**x*a + b)*a**2 + 2*e**(2*x)*log(e**(2*x)*b + 2*e**x*a + b)*b**2 - 2*e* 
*x*a*b + log(e**(2*x) + 1)*a**2 - log(e**(2*x) + 1)*b**2 - log(e**(2*x)*b 
+ 2*e**x*a + b)*a**2 + log(e**(2*x)*b + 2*e**x*a + b)*b**2 - a**2)/(a**3*( 
e**(4*x) + 2*e**(2*x) + 1))