3.212 \(\int \text {sech}(3 x) \sinh (x) \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{6} \log \left (3-4 \cosh ^2(x)\right )-\frac {1}{3} \log (\cosh (x)) \]

[Out]

-1/3*ln(cosh(x))+1/6*ln(3-4*cosh(x)^2)

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4357, 266, 36, 29, 31} \[ \frac {1}{6} \log \left (3-4 \cosh ^2(x)\right )-\frac {1}{3} \log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sech[3*x]*Sinh[x],x]

[Out]

-Log[Cosh[x]]/3 + Log[3 - 4*Cosh[x]^2]/6

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]

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4357

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rubi steps

\begin {align*} \int \text {sech}(3 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (-3+4 x^2\right )} \, dx,x,\cosh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (-3+4 x)} \, dx,x,\cosh ^2(x)\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\cosh ^2(x)\right )\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-3+4 x} \, dx,x,\cosh ^2(x)\right )\\ &=-\frac {1}{3} \log (\cosh (x))+\frac {1}{6} \log \left (3-4 \cosh ^2(x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.81 \[ -\frac {1}{3} \tanh ^{-1}\left (\frac {1}{3} \left (8 \sinh ^2(x)+5\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[3*x]*Sinh[x],x]

[Out]

-1/3*ArcTanh[(5 + 8*Sinh[x]^2)/3]

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fricas [B]  time = 0.43, size = 52, normalized size = 2.48 \[ \frac {1}{6} \, \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(3*x)*sinh(x),x, algorithm="fricas")

[Out]

1/6*log((2*cosh(x)^2 + 2*sinh(x)^2 - 1)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 1/3*log(2*cosh(x)/(cosh
(x) - sinh(x)))

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giac [B]  time = 0.11, size = 41, normalized size = 1.95 \[ \frac {1}{6} \, \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(3*x)*sinh(x),x, algorithm="giac")

[Out]

1/6*log(sqrt(3)*e^x + e^(2*x) + 1) + 1/6*log(-sqrt(3)*e^x + e^(2*x) + 1) - 1/3*log(e^(2*x) + 1)

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maple [A]  time = 0.21, size = 26, normalized size = 1.24 \[ -\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{3}+\frac {\ln \left (1-{\mathrm e}^{2 x}+{\mathrm e}^{4 x}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(3*x)*sinh(x),x)

[Out]

-1/3*ln(1+exp(2*x))+1/6*ln(1-exp(2*x)+exp(4*x))

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maxima [B]  time = 0.44, size = 45, normalized size = 2.14 \[ \frac {1}{6} \, \log \left (\sqrt {3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{6} \, \log \left (-\sqrt {3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(3*x)*sinh(x),x, algorithm="maxima")

[Out]

1/6*log(sqrt(3)*e^(-x) + e^(-2*x) + 1) + 1/6*log(-sqrt(3)*e^(-x) + e^(-2*x) + 1) - 1/3*log(e^(-2*x) + 1)

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mupad [B]  time = 1.46, size = 27, normalized size = 1.29 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}-1\right )}{6}-\frac {\ln \left (3\,{\mathrm {e}}^{2\,x}+3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/cosh(3*x),x)

[Out]

log(exp(2*x) - exp(4*x) - 1)/6 - log(3*exp(2*x) + 3)/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (3 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(3*x)*sinh(x),x)

[Out]

Integral(sinh(x)*sech(3*x), x)

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