3.76 \(\int \frac{\tanh (x)}{1+\cosh ^2(x)} \, dx\)

Optimal. Leaf size=15 \[ \log (\cosh (x))-\frac{1}{2} \log \left (\cosh ^2(x)+1\right ) \]

[Out]

Log[Cosh[x]] - Log[1 + Cosh[x]^2]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0326715, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3194, 36, 29, 31} \[ \log (\cosh (x))-\frac{1}{2} \log \left (\cosh ^2(x)+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/(1 + Cosh[x]^2),x]

[Out]

Log[Cosh[x]] - Log[1 + Cosh[x]^2]/2

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\tanh (x)}{1+\cosh ^2(x)} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\cosh ^2(x)\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\cosh ^2(x)\right )\\ &=\log (\cosh (x))-\frac{1}{2} \log \left (1+\cosh ^2(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0082814, size = 15, normalized size = 1. \[ \log (\cosh (x))-\frac{1}{2} \log \left (\cosh ^2(x)+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]/(1 + Cosh[x]^2),x]

[Out]

Log[Cosh[x]] - Log[1 + Cosh[x]^2]/2

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( \cosh \left ( x \right ) \right ) -{\frac{\ln \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)/(1+cosh(x)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10785, size = 31, normalized size = 2.07 \begin{align*} -\frac{1}{2} \, \log \left (6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1\right ) + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)/(1+cosh(x)^2),x, algorithm="maxima")

[Out]

-1/2*log(6*e^(-2*x) + e^(-4*x) + 1) + log(e^(-2*x) + 1)

________________________________________________________________________________________

Fricas [B]  time = 2.57228, size = 165, normalized size = 11. \begin{align*} -\frac{1}{2} \, \log \left (\frac{2 \,{\left (\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)/(1+cosh(x)^2),x, algorithm="fricas")

[Out]

-1/2*log(2*(cosh(x)^2 + sinh(x)^2 + 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + log(2*cosh(x)/(cosh(x) -
sinh(x)))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh{\left (x \right )}}{\cosh ^{2}{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)/(1+cosh(x)**2),x)

[Out]

Integral(tanh(x)/(cosh(x)**2 + 1), x)

________________________________________________________________________________________

Giac [A]  time = 1.23053, size = 31, normalized size = 2.07 \begin{align*} -\frac{1}{2} \, \log \left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)/(1+cosh(x)^2),x, algorithm="giac")

[Out]

-1/2*log(e^(4*x) + 6*e^(2*x) + 1) + log(e^(2*x) + 1)