3.215 $$\int \text{sech}(6 x) \sinh (x) \, dx$$

Optimal. Leaf size=85 $\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}}$

[Out]

ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) - ArcTanh[
(2*Cosh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])

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Rubi [A]  time = 0.0791248, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.571, Rules used = {4357, 2057, 207, 1166} $\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[6*x]*Sinh[x],x]

[Out]

ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) - ArcTanh[
(2*Cosh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])

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 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1166

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

Rubi steps

\begin{align*} \int \text{sech}(6 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{-1+18 x^2-48 x^4+32 x^6} \, dx,x,\cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{3 \left (-1+2 x^2\right )}+\frac{4 \left (-1+2 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{-1+2 x^2}{1-16 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-8-4 \sqrt{3}+16 x^2} \, dx,x,\cosh (x)\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-8+4 \sqrt{3}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{3}}}\right )}{6 \sqrt{2-\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{3}}}\right )}{6 \sqrt{2+\sqrt{3}}}\\ \end{align*}

Mathematica [C]  time = 0.26666, size = 385, normalized size = 4.53 $\frac{\sqrt{2} \text{RootSum}\left [\text{\#1}^8-\text{\#1}^4+1\& ,\frac{\text{\#1}^6 x-\text{\#1}^4 x+\text{\#1}^2 x+2 \text{\#1}^6 \log \left (-\text{\#1} \sinh \left (\frac{x}{2}\right )+\text{\#1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \text{\#1}^4 \log \left (-\text{\#1} \sinh \left (\frac{x}{2}\right )+\text{\#1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )+2 \text{\#1}^2 \log \left (-\text{\#1} \sinh \left (\frac{x}{2}\right )+\text{\#1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \log \left (-\text{\#1} \sinh \left (\frac{x}{2}\right )+\text{\#1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-x}{2 \text{\#1}^7-\text{\#1}^3}\& \right ]+8 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )-2 \log \left (\sqrt{2}-2 \cosh (x)\right )+2 \log \left (2 \cosh (x)+\sqrt{2}\right )+4 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )-4 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{24 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[6*x]*Sinh[x],x]

[Out]

((4*I)*ArcTan[(Cosh[x/2] + Sinh[x/2])/((1 + Sqrt[2])*Cosh[x/2] - (-1 + Sqrt[2])*Sinh[x/2])] - (4*I)*ArcTan[(Co
sh[x/2] + Sinh[x/2])/((-1 + Sqrt[2])*Cosh[x/2] - (1 + Sqrt[2])*Sinh[x/2])] + 8*ArcTanh[Sqrt[2] - I*Tanh[x/2]]
- 2*Log[Sqrt[2] - 2*Cosh[x]] + 2*Log[Sqrt[2] + 2*Cosh[x]] + Sqrt[2]*RootSum[1 - #1^4 + #1^8 & , (-x - 2*Log[-C
osh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + x*#1^2 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - S
inh[x/2]*#1]*#1^2 - x*#1^4 - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 + x*#1^6 + 2*Log
[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-#1^3 + 2*#1^7) & ])/(24*Sqrt[2])

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Maple [C]  time = 0.068, size = 78, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{12}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{12}}+2\,\sum _{{\it \_R}={\it RootOf} \left ( 331776\,{{\it \_Z}}^{4}-2304\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{2\,x}}+ \left ( 13824\,{{\it \_R}}^{3}-96\,{\it \_R} \right ){{\rm e}^{x}}+1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(6*x)*sinh(x),x)

[Out]

1/12*ln(1+exp(2*x)+exp(x)*2^(1/2))*2^(1/2)-1/12*ln(1+exp(2*x)-exp(x)*2^(1/2))*2^(1/2)+2*sum(_R*ln(exp(2*x)+(13
824*_R^3-96*_R)*exp(x)+1),_R=RootOf(331776*_Z^4-2304*_Z^2+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{12} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \int \frac{e^{\left (7 \, x\right )} - e^{\left (5 \, x\right )} + e^{\left (3 \, x\right )} - e^{x}}{3 \,{\left (e^{\left (8 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x, algorithm="maxima")

[Out]

1/12*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) - 1/12*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) + integrate(1/3*(e^
(7*x) - e^(5*x) + e^(3*x) - e^x)/(e^(8*x) - e^(4*x) + 1), x)

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Fricas [B]  time = 2.33375, size = 906, normalized size = 10.66 \begin{align*} \frac{1}{12} \, \sqrt{\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} +{\left ({\left (\sqrt{3} - 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} - 2\right )} \sinh \left (x\right )\right )} \sqrt{\sqrt{3} + 2} + 1\right ) - \frac{1}{12} \, \sqrt{\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} -{\left ({\left (\sqrt{3} - 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} - 2\right )} \sinh \left (x\right )\right )} \sqrt{\sqrt{3} + 2} + 1\right ) - \frac{1}{12} \, \sqrt{-\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} +{\left ({\left (\sqrt{3} + 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} + 2\right )} \sinh \left (x\right )\right )} \sqrt{-\sqrt{3} + 2} + 1\right ) + \frac{1}{12} \, \sqrt{-\sqrt{3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} -{\left ({\left (\sqrt{3} + 2\right )} \cosh \left (x\right ) +{\left (\sqrt{3} + 2\right )} \sinh \left (x\right )\right )} \sqrt{-\sqrt{3} + 2} + 1\right ) + \frac{1}{12} \, \sqrt{2} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x, algorithm="fricas")

[Out]

1/12*sqrt(sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(3) - 2)*cosh(x) + (sqrt(3) - 2)*
sinh(x))*sqrt(sqrt(3) + 2) + 1) - 1/12*sqrt(sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqr
t(3) - 2)*cosh(x) + (sqrt(3) - 2)*sinh(x))*sqrt(sqrt(3) + 2) + 1) - 1/12*sqrt(-sqrt(3) + 2)*log(cosh(x)^2 + 2*
cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)*sinh(x))*sqrt(-sqrt(3) + 2) + 1) + 1/12*s
qrt(-sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)*sinh(
x))*sqrt(-sqrt(3) + 2) + 1) + 1/12*sqrt(2)*log((cosh(x)^2 + sinh(x)^2 + 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + si
nh(x)^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{sech}{\left (6 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x)

[Out]

Integral(sinh(x)*sech(6*x), x)

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Giac [B]  time = 1.24073, size = 208, normalized size = 2.45 \begin{align*} -\frac{1}{24} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (\frac{1}{2} \,{\left (\sqrt{6} + \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{24} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (-\frac{1}{2} \,{\left (\sqrt{6} + \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{24} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (\frac{1}{2} \,{\left (\sqrt{6} - \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{24} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (-\frac{1}{2} \,{\left (\sqrt{6} - \sqrt{2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{12} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x, algorithm="giac")

[Out]

-1/24*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*e^x + e^(2*x) + 1) + 1/24*(sqrt(6) - sqrt(2))*log(-1/2*(
sqrt(6) + sqrt(2))*e^x + e^(2*x) + 1) - 1/24*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) - sqrt(2))*e^x + e^(2*x) + 1
) + 1/24*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*e^x + e^(2*x) + 1) + 1/12*sqrt(2)*log(sqrt(2)*e^x +
e^(2*x) + 1) - 1/12*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1)