\(\int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx\) [117]

Optimal result

Integrand size = 13, antiderivative size = 35 \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}+\frac {\left (1-\frac {a^2}{b^2}\right ) \log (a+b \text {sech}(x))}{a}+\frac {\text {sech}(x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3970, 908} \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (1-\frac {a^2}{b^2}\right ) \log (a+b \text {sech}(x))}{a}+\frac {\log (\cosh (x))}{a}+\frac {\text {sech}(x)}{b} \]

[In]

Int[Tanh[x]^3/(a + b*Sech[x]),x]

[Out]

Log[Cosh[x]]/a + ((1 - a^2/b^2)*Log[a + b*Sech[x]])/a + Sech[x]/b

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {b^2-x^2}{x (a+x)} \, dx,x,b \text {sech}(x)\right )}{b^2} \\ & = -\frac {\text {Subst}\left (\int \left (-1+\frac {b^2}{a x}+\frac {a^2-b^2}{a (a+x)}\right ) \, dx,x,b \text {sech}(x)\right )}{b^2} \\ & = \frac {\log (\cosh (x))}{a}+\frac {\left (1-\frac {a^2}{b^2}\right ) \log (a+b \text {sech}(x))}{a}+\frac {\text {sech}(x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}+\frac {\log (a+b \text {sech}(x))}{a}-\frac {a \log (a+b \text {sech}(x))}{b^2}+\frac {\text {sech}(x)}{b} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(35)=70\).

Time = 0.39 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.14

method	result	size
risch	\(-\frac {x}{a}+\frac {2 \,{\mathrm e}^{x}}{b \left (1+{\mathrm e}^{2 x}\right )}+\frac {a \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a}\)	\(75\)
default	\(\frac {\frac {2 b}{1+\tanh \left (\frac {x}{2}\right )^{2}}+a \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{b^{2}}-\frac {\left (a -b \right ) \left (a +b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a \,b^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\)	\(92\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (35) = 70\).

Time = 0.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 5.71 \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {b^{2} x \cosh \left (x\right )^{2} + b^{2} x \sinh \left (x\right )^{2} + b^{2} x - 2 \, a b \cosh \left (x\right ) + {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (b^{2} x \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )}{a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} + a b^{2}} \]

[In]

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

[Out]

-(b^2*x*cosh(x)^2 + b^2*x*sinh(x)^2 + b^2*x - 2*a*b*cosh(x) + ((a^2 - b^2)*cosh(x)^2 + 2*(a^2 - b^2)*cosh(x)*s
inh(x) + (a^2 - b^2)*sinh(x)^2 + a^2 - b^2)*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) - (a^2*cosh(x)^2 + 2*a^
2*cosh(x)*sinh(x) + a^2*sinh(x)^2 + a^2)*log(2*cosh(x)/(cosh(x) - sinh(x))) + 2*(b^2*x*cosh(x) - a*b)*sinh(x))
/(a*b^2*cosh(x)^2 + 2*a*b^2*cosh(x)*sinh(x) + a*b^2*sinh(x)^2 + a*b^2)

Sympy [F]

\[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91 \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {x}{a} + \frac {2 \, e^{\left (-x\right )}}{b e^{\left (-2 \, x\right )} + b} + \frac {a \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a b^{2}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35) = 70\).

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.09 \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {a \log \left (e^{\left (-x\right )} + e^{x}\right )}{b^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a b^{2}} - \frac {a {\left (e^{\left (-x\right )} + e^{x}\right )} - 2 \, b}{b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 7.43 \[ \int \frac {\tanh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {2\,{\mathrm {e}}^x}{b+b\,{\mathrm {e}}^{2\,x}}-\frac {x}{a}+\frac {\ln \left (16\,a^5\,{\mathrm {e}}^{2\,x}+4\,a\,b^4+16\,a^5-16\,a^3\,b^2+8\,b^5\,{\mathrm {e}}^x-16\,a^3\,b^2\,{\mathrm {e}}^{2\,x}+32\,a^4\,b\,{\mathrm {e}}^x+4\,a\,b^4\,{\mathrm {e}}^{2\,x}-32\,a^2\,b^3\,{\mathrm {e}}^x\right )}{a}-\frac {a\,\ln \left (16\,a^5\,{\mathrm {e}}^{2\,x}+4\,a\,b^4+16\,a^5-16\,a^3\,b^2+8\,b^5\,{\mathrm {e}}^x-16\,a^3\,b^2\,{\mathrm {e}}^{2\,x}+32\,a^4\,b\,{\mathrm {e}}^x+4\,a\,b^4\,{\mathrm {e}}^{2\,x}-32\,a^2\,b^3\,{\mathrm {e}}^x\right )}{b^2}+\frac {a\,\ln \left (16\,a^6\,{\mathrm {e}}^{2\,x}-4\,b^6\,{\mathrm {e}}^{2\,x}+16\,a^6-4\,b^6+20\,a^2\,b^4-32\,a^4\,b^2+20\,a^2\,b^4\,{\mathrm {e}}^{2\,x}-32\,a^4\,b^2\,{\mathrm {e}}^{2\,x}\right )}{b^2} \]

[In]

int(tanh(x)^3/(a + b/cosh(x)),x)

[Out]

(2*exp(x))/(b + b*exp(2*x)) - x/a + log(16*a^5*exp(2*x) + 4*a*b^4 + 16*a^5 - 16*a^3*b^2 + 8*b^5*exp(x) - 16*a^
3*b^2*exp(2*x) + 32*a^4*b*exp(x) + 4*a*b^4*exp(2*x) - 32*a^2*b^3*exp(x))/a - (a*log(16*a^5*exp(2*x) + 4*a*b^4
+ 16*a^5 - 16*a^3*b^2 + 8*b^5*exp(x) - 16*a^3*b^2*exp(2*x) + 32*a^4*b*exp(x) + 4*a*b^4*exp(2*x) - 32*a^2*b^3*e
xp(x)))/b^2 + (a*log(16*a^6*exp(2*x) - 4*b^6*exp(2*x) + 16*a^6 - 4*b^6 + 20*a^2*b^4 - 32*a^4*b^2 + 20*a^2*b^4*
exp(2*x) - 32*a^4*b^2*exp(2*x)))/b^2