3.1.1 \(\int \frac {2}{-1+3 \cosh (4+6 x)} \, dx\) [1]

Optimal. Leaf size=22 \[ \frac {\text {ArcTan}\left (\sqrt {2} \tanh (2+3 x)\right )}{3 \sqrt {2}} \]

[Out]

1/6*arctan(2^(1/2)*tanh(2+3*x))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 2738, 212} \begin {gather*} \frac {\text {ArcTan}\left (\sqrt {2} \tanh (3 x+2)\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2/(-1 + 3*Cosh[4 + 6*x]),x]

[Out]

ArcTan[Sqrt[2]*Tanh[2 + 3*x]]/(3*Sqrt[2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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

Rule 2738

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

Rubi steps

\begin {align*} \int \frac {2}{-1+3 \cosh (4+6 x)} \, dx &=2 \int \frac {1}{-1+3 \cosh (4+6 x)} \, dx\\ &=-\left (\frac {2}{3} i \text {Subst}\left (\int \frac {1}{2-4 x^2} \, dx,x,\tan \left (\frac {1}{2} (4 i+6 i x)\right )\right )\right )\\ &=\frac {\tan ^{-1}\left (\sqrt {2} \tanh (2+3 x)\right )}{3 \sqrt {2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
time = 0.06, size = 47, normalized size = 2.14 \begin {gather*} \frac {\text {ArcTan}\left (\frac {3 \left (-1+e^8\right )+\left (3+2 e^4+3 e^8\right ) \tanh (3 x)}{4 \sqrt {2} e^4}\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2/(-1 + 3*Cosh[4 + 6*x]),x]

[Out]

ArcTan[(3*(-1 + E^8) + (3 + 2*E^4 + 3*E^8)*Tanh[3*x])/(4*Sqrt[2]*E^4)]/(3*Sqrt[2])

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Maple [A]
time = 1.09, size = 17, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {\arctan \left (\sqrt {2}\, \tanh \left (2+3 x \right )\right ) \sqrt {2}}{6}\)	\(17\)
default	\(\frac {\arctan \left (\sqrt {2}\, \tanh \left (2+3 x \right )\right ) \sqrt {2}}{6}\)	\(17\)
risch	\(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{4+6 x}-\frac {1}{3}+\frac {2 i \sqrt {2}}{3}\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{4+6 x}-\frac {1}{3}-\frac {2 i \sqrt {2}}{3}\right )}{12}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(-1+3*cosh(4+6*x)),x,method=_RETURNVERBOSE)

[Out]

1/6*arctan(2^(1/2)*tanh(2+3*x))*2^(1/2)

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Maxima [A]
time = 0.48, size = 21, normalized size = 0.95 \begin {gather*} -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (-6 \, x - 4\right )} - 1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-1+3*cosh(4+6*x)),x, algorithm="maxima")

[Out]

-1/6*sqrt(2)*arctan(1/4*sqrt(2)*(3*e^(-6*x - 4) - 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).
time = 0.37, size = 34, normalized size = 1.55 \begin {gather*} \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {3}{4} \, \sqrt {2} \cosh \left (6 \, x + 4\right ) + \frac {3}{4} \, \sqrt {2} \sinh \left (6 \, x + 4\right ) - \frac {1}{4} \, \sqrt {2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-1+3*cosh(4+6*x)),x, algorithm="fricas")

[Out]

1/6*sqrt(2)*arctan(3/4*sqrt(2)*cosh(6*x + 4) + 3/4*sqrt(2)*sinh(6*x + 4) - 1/4*sqrt(2))

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Sympy [A]
time = 0.12, size = 19, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \tanh {\left (3 x + 2 \right )} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-1+3*cosh(4+6*x)),x)

[Out]

sqrt(2)*atan(sqrt(2)*tanh(3*x + 2))/6

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Giac [A]
time = 0.38, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (6 \, x + 4\right )} - 1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(-1+3*cosh(4+6*x)),x, algorithm="giac")

[Out]

1/6*sqrt(2)*arctan(1/4*sqrt(2)*(3*e^(6*x + 4) - 1))

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Mupad [B]
time = 0.12, size = 21, normalized size = 0.95 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )}{4}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(3*cosh(6*x + 4) - 1),x)

[Out]

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

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