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]

ln(cosh(x))/a+sech(x)/a

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Rubi [A]  time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [B]  time = 0.40, size = 85, normalized size = 6.07 \[ -\frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (x \cosh \relax (x) - 1\right )} \sinh \relax (x) + x - 2 \, \cosh \relax (x)}{a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} + a} \]

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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giac [B]  time = 0.12, size = 35, normalized size = 2.50 \[ \frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {e^{\left (-x\right )} + e^{x} - 2}{a {\left (e^{\left (-x\right )} + e^{x}\right )}} \]

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))

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maple [B]  time = 0.13, size = 54, normalized size = 3.86 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {2}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{a} \]

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

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maxima [B]  time = 0.71, size = 33, normalized size = 2.36 \[ \frac {x}{a} + \frac {2 \, e^{\left (-x\right )}}{a e^{\left (-2 \, x\right )} + a} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]

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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mupad [B]  time = 1.36, size = 33, normalized size = 2.36 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{a}-\frac {x}{a}+\frac {2\,{\mathrm {e}}^x}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^3/(a + a/cosh(x)),x)

[Out]

log(exp(2*x) + 1)/a - x/a + (2*exp(x))/(a*(exp(2*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tanh ^{3}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]

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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