3.106 \(\int \frac{\tanh ^3(x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=14 \[ \frac{\text{sech}(x)}{a}+\frac{\log (\cosh (x))}{a} \]

[Out]

Log[Cosh[x]]/a + Sech[x]/a

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Rubi [A]  time = 0.0483187, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3879, 43} \[ \frac{\text{sech}(x)}{a}+\frac{\log (\cosh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^3/(a + a*Sech[x]),x]

[Out]

Log[Cosh[x]]/a + Sech[x]/a

Rule 3879

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\tanh ^3(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x^2} \, dx,x,\cosh (x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{a}{x}\right ) \, dx,x,\cosh (x)\right )}{a^2}\\ &=\frac{\log (\cosh (x))}{a}+\frac{\text{sech}(x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0335871, size = 10, normalized size = 0.71 \[ \frac{\text{sech}(x)+\log (\cosh (x))}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[Cosh[x]] + Sech[x])/a

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Maple [B]  time = 0.027, size = 54, normalized size = 3.9 \begin{align*} -{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{1}{a}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }+2\,{\frac{1}{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.65543, size = 45, normalized size = 3.21 \begin{align*} \frac{x}{a} + \frac{2 \, e^{\left (-x\right )}}{a e^{\left (-2 \, x\right )} + a} + \frac{\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.52471, size = 288, normalized size = 20.57 \begin{align*} -\frac{x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (x \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + x - 2 \, \cosh \left (x\right )}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tanh ^{3}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(x)**3/(sech(x) + 1), x)/a

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Giac [B]  time = 1.22351, size = 47, normalized size = 3.36 \begin{align*} \frac{\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac{e^{\left (-x\right )} + e^{x} - 2}{a{\left (e^{\left (-x\right )} + e^{x}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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