### 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]

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

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Rubi [A]  time = 0.0301864, antiderivative size = 21, normalized size of antiderivative = 1., 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 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])

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 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 \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*}

Mathematica [A]  time = 0.0086202, 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]

-ArcTanh[(5 + 8*Sinh[x]^2)/3]/3

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Maple [A]  time = 0.034, size = 26, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({{\rm e}^{2\,x}}+1 \right ) }{3}}+{\frac{\ln \left ({{\rm e}^{4\,x}}-{{\rm e}^{2\,x}}+1 \right ) }{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.56583, size = 61, normalized size = 2.9 \begin{align*} \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 ) \end{align*}

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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Fricas [B]  time = 2.05395, size = 171, normalized size = 8.14 \begin{align*} \frac{1}{6} \, \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - \frac{1}{3} \, \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(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{sech}{\left (3 x \right )}\, dx \end{align*}

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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Giac [B]  time = 1.15759, size = 55, normalized size = 2.62 \begin{align*} \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 ) \end{align*}

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)