3.3.42 \(\int \cosh (x) \text {sech}(4 x) \, dx\) [242]

Optimal. Leaf size=71 \[ \frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \]

[Out]

1/2*arctan(2*sinh(x)/(2-2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/2*arctan(2*sinh(x)/(2+2^(1/2))^(1/2))/(4+2*2^(1/
2))^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4441, 1107, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Sech[4*x],x]

[Out]

ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqr
t[2*(2 + Sqrt[2])])

Rule 209

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

Rule 1107

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

Rule 4441

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

Rubi steps

\begin {align*} \int \cosh (x) \text {sech}(4 x) \, dx &=\text {Subst}\left (\int \frac {1}{1+8 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=\sqrt {2} \text {Subst}\left (\int \frac {1}{4-2 \sqrt {2}+8 x^2} \, dx,x,\sinh (x)\right )-\sqrt {2} \text {Subst}\left (\int \frac {1}{4+2 \sqrt {2}+8 x^2} \, dx,x,\sinh (x)\right )\\ &=\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.02, size = 108, normalized size = 1.52 \begin {gather*} \frac {1}{16} \text {RootSum}\left [1+\text {$\#$1}^8\&,\frac {x+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2}{\text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Sech[4*x],x]

[Out]

RootSum[1 + #1^8 & , (x + 2*Log[-Cosh[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 - Sinh[x/2]*#1]*#1^2)/#1^5 & ]/16

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.14, size = 40, normalized size = 0.56

method	result	size
risch	\(2 \left (\munderset {\textit {\_R} =\RootOf \left (32768 \textit {\_Z}^{4}+512 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-4096 \textit {\_R}^{3}-48 \textit {\_R} \right ) {\mathrm e}^{x}-1\right )\right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sech(4*x),x,method=_RETURNVERBOSE)

[Out]

2*sum(_R*ln(exp(2*x)+(-4096*_R^3-48*_R)*exp(x)-1),_R=RootOf(32768*_Z^4+512*_Z^2+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(4*x),x, algorithm="maxima")

[Out]

integrate(cosh(x)*sech(4*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (49) = 98\).
time = 0.40, size = 142, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {1}{2} \, {\left ({\left (\sqrt {2} e^{\left (2 \, x\right )} - \sqrt {2}\right )} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt {\sqrt {2} + 2}\right )} e^{\left (-x\right )}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {1}{2} \, {\left ({\left (\sqrt {2} e^{\left (2 \, x\right )} - \sqrt {2}\right )} \sqrt {-\sqrt {2} + 2} - \sqrt {2} \sqrt {\sqrt {2} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt {-\sqrt {2} + 2}\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(4*x),x, algorithm="fricas")

[Out]

-1/2*sqrt(sqrt(2) + 2)*arctan(-1/2*((sqrt(2)*e^(2*x) - sqrt(2))*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2)*e^(2
*x) + e^(4*x) + 1)*sqrt(sqrt(2) + 2))*e^(-x)) + 1/2*sqrt(-sqrt(2) + 2)*arctan(-1/2*((sqrt(2)*e^(2*x) - sqrt(2)
)*sqrt(-sqrt(2) + 2) - sqrt(2)*sqrt(sqrt(2)*e^(2*x) + e^(4*x) + 1)*sqrt(-sqrt(2) + 2))*e^(-x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh {\left (x \right )} \operatorname {sech}{\left (4 x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(4*x),x)

[Out]

Integral(cosh(x)*sech(4*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (49) = 98\).
time = 0.43, size = 135, normalized size = 1.90 \begin {gather*} \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(4*x),x, algorithm="giac")

[Out]

1/4*sqrt(sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*e^x)/sqrt(-sqrt(2) + 2)) + 1/4*sqrt(sqrt(2) + 2)*arctan(-(
sqrt(sqrt(2) + 2) - 2*e^x)/sqrt(-sqrt(2) + 2)) - 1/4*sqrt(-sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*e^x)/sq
rt(sqrt(2) + 2)) - 1/4*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*e^x)/sqrt(sqrt(2) + 2))

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Mupad [B]
time = 2.95, size = 126, normalized size = 1.77 \begin {gather*} \frac {\mathrm {atan}\left (\frac {3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3}{{\mathrm {e}}^x\,\sqrt {\sqrt {2}+2}+\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\sqrt {2}+2}}\right )\,\sqrt {\sqrt {2}+2}}{4}+\frac {\mathrm {atan}\left (\frac {3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3}{{\mathrm {e}}^x\,\sqrt {2-\sqrt {2}}-\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {2-\sqrt {2}}}\right )\,\sqrt {2-\sqrt {2}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/cosh(4*x),x)

[Out]

(atan((3*exp(2*x) - 2*2^(1/2) + 2*2^(1/2)*exp(2*x) - 3)/(exp(x)*(2^(1/2) + 2)^(1/2) + 2^(1/2)*exp(x)*(2^(1/2)
+ 2)^(1/2)))*(2^(1/2) + 2)^(1/2))/4 + (atan((3*exp(2*x) + 2*2^(1/2) - 2*2^(1/2)*exp(2*x) - 3)/(exp(x)*(2 - 2^(
1/2))^(1/2) - 2^(1/2)*exp(x)*(2 - 2^(1/2))^(1/2)))*(2 - 2^(1/2))^(1/2))/4

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