### 3.214 $$\int \text{sech}(5 x) \sinh (x) \, dx$$

Optimal. Leaf size=62 $-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)-\sqrt{5}+5\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)+\sqrt{5}+5\right )+\frac{1}{5} \log (\cosh (x))$

[Out]

Log[Cosh[x]]/5 - ((1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Cosh[x]^2])/20 - ((1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Cosh[x
]^2])/20

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Rubi [A]  time = 0.0833591, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.857, Rules used = {4357, 1114, 705, 29, 632, 31} $-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)-\sqrt{5}+5\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)+\sqrt{5}+5\right )+\frac{1}{5} \log (\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[5*x]*Sinh[x],x]

[Out]

Log[Cosh[x]]/5 - ((1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Cosh[x]^2])/20 - ((1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Cosh[x
]^2])/20

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 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 29

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

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

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}(5 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (5-20 x^2+16 x^4\right )} \, dx,x,\cosh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \left (5-20 x+16 x^2\right )} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\cosh ^2(x)\right )+\frac{1}{10} \operatorname{Subst}\left (\int \frac{20-16 x}{5-20 x+16 x^2} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{5} \log (\cosh (x))-\frac{1}{5} \left (4 \left (1-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10-2 \sqrt{5}+16 x} \, dx,x,\cosh ^2(x)\right )-\frac{1}{5} \left (4 \left (1+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10+2 \sqrt{5}+16 x} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{5} \log (\cosh (x))-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (5-\sqrt{5}-8 \cosh ^2(x)\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (5+\sqrt{5}-8 \cosh ^2(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.083173, size = 57, normalized size = 0.92 $\frac{1}{20} \left (\left (\sqrt{5}-1\right ) \log \left (8 \sinh ^2(x)-\sqrt{5}+3\right )-\left (1+\sqrt{5}\right ) \log \left (8 \sinh ^2(x)+\sqrt{5}+3\right )+4 \log (\cosh (x))\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[5*x]*Sinh[x],x]

[Out]

(4*Log[Cosh[x]] + (-1 + Sqrt[5])*Log[3 - Sqrt[5] + 8*Sinh[x]^2] - (1 + Sqrt[5])*Log[3 + Sqrt[5] + 8*Sinh[x]^2]
)/20

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Maple [B]  time = 0.058, size = 101, normalized size = 1.6 \begin{align*}{\frac{\ln \left ({{\rm e}^{2\,x}}+1 \right ) }{5}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) }{20}}+{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) }{20}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) \sqrt{5}}{20}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(5*x)*sinh(x),x)

[Out]

1/5*ln(exp(2*x)+1)-1/20*ln(exp(4*x)+(-1/2-1/2*5^(1/2))*exp(2*x)+1)+1/20*ln(exp(4*x)+(-1/2-1/2*5^(1/2))*exp(2*x
)+1)*5^(1/2)-1/20*ln(exp(4*x)+(1/2*5^(1/2)-1/2)*exp(2*x)+1)-1/20*ln(exp(4*x)+(1/2*5^(1/2)-1/2)*exp(2*x)+1)*5^(
1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{5} \, \int \frac{{\left (e^{\left (6 \, x\right )} - e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (2 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac{2}{5} \, \int \frac{e^{\left (6 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac{1}{5} \, \int \frac{e^{\left (4 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} - \frac{4}{5} \, \int \frac{e^{\left (2 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac{1}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(5*x)*sinh(x),x, algorithm="maxima")

[Out]

-2/5*integrate((e^(6*x) - e^(4*x) + e^(2*x) - 1)*e^(2*x)/(e^(8*x) - e^(6*x) + e^(4*x) - e^(2*x) + 1), x) + 2/5
*integrate(e^(6*x)/(e^(8*x) - e^(6*x) + e^(4*x) - e^(2*x) + 1), x) + 1/5*integrate(e^(4*x)/(e^(8*x) - e^(6*x)
+ e^(4*x) - e^(2*x) + 1), x) - 4/5*integrate(e^(2*x)/(e^(8*x) - e^(6*x) + e^(4*x) - e^(2*x) + 1), x) + 1/5*log
(e^(2*x) + 1)

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Fricas [B]  time = 2.19283, size = 583, normalized size = 9.4 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{4 \, \cosh \left (x\right )^{4} + 4 \, \sinh \left (x\right )^{4} - 4 \,{\left (\sqrt{5} + 1\right )} \cosh \left (x\right )^{2} + 4 \,{\left (6 \, \cosh \left (x\right )^{2} - \sqrt{5} - 1\right )} \sinh \left (x\right )^{2} + \sqrt{5} + 7}{2 \, \cosh \left (x\right )^{4} + 2 \, \sinh \left (x\right )^{4} + 2 \,{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 1}\right ) - \frac{1}{20} \, \log \left (\frac{2 \, \cosh \left (x\right )^{4} + 2 \, \sinh \left (x\right )^{4} + 2 \,{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}\right ) + \frac{1}{5} \, \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(5*x)*sinh(x),x, algorithm="fricas")

[Out]

1/20*sqrt(5)*log((4*cosh(x)^4 + 4*sinh(x)^4 - 4*(sqrt(5) + 1)*cosh(x)^2 + 4*(6*cosh(x)^2 - sqrt(5) - 1)*sinh(x
)^2 + sqrt(5) + 7)/(2*cosh(x)^4 + 2*sinh(x)^4 + 2*(6*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 1)) - 1/20*log((
2*cosh(x)^4 + 2*sinh(x)^4 + 2*(6*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 1)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x)
+ 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) + 1/5*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 (5 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(5*x)*sinh(x),x)

[Out]

Integral(sinh(x)*sech(5*x), x)

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Giac [B]  time = 1.20215, size = 159, normalized size = 2.56 \begin{align*} \frac{1}{20} \,{\left (\sqrt{5} - 1\right )} \log \left (\frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{20} \,{\left (\sqrt{5} - 1\right )} \log \left (-\frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{20} \,{\left (\sqrt{5} + 1\right )} \log \left (\frac{1}{2} \, \sqrt{-2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{20} \,{\left (\sqrt{5} + 1\right )} \log \left (-\frac{1}{2} \, \sqrt{-2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(5*x)*sinh(x),x, algorithm="giac")

[Out]

1/20*(sqrt(5) - 1)*log(1/2*sqrt(2*sqrt(5) + 10)*e^x + e^(2*x) + 1) + 1/20*(sqrt(5) - 1)*log(-1/2*sqrt(2*sqrt(5
) + 10)*e^x + e^(2*x) + 1) - 1/20*(sqrt(5) + 1)*log(1/2*sqrt(-2*sqrt(5) + 10)*e^x + e^(2*x) + 1) - 1/20*(sqrt(
5) + 1)*log(-1/2*sqrt(-2*sqrt(5) + 10)*e^x + e^(2*x) + 1) + 1/5*log(e^(2*x) + 1)