3.2 \(\int \frac {1}{\cosh ^2(2+3 x)+2 \sinh ^2(2+3 x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {\tan ^{-1}\left (\sqrt {2} \tanh (3 x+2)\right )}{3 \sqrt {2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {203} \[ \frac {\tan ^{-1}\left (\sqrt {2} \tanh (3 x+2)\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[2 + 3*x]^2 + 2*Sinh[2 + 3*x]^2)^(-1),x]

[Out]

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

Rule 203

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

Rubi steps

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

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Mathematica [B]  time = 0.07, size = 47, normalized size = 2.14 \[ \frac {\tan ^{-1}\left (\frac {\left (3+2 e^4+3 e^8\right ) \tanh (3 x)+3 \left (e^8-1\right )}{4 \sqrt {2} e^4}\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[2 + 3*x]^2 + 2*Sinh[2 + 3*x]^2)^(-1),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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fricas [B]  time = 0.42, size = 47, normalized size = 2.14 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \left (3 \, x + 2\right ) + 2 \, \sqrt {2} \sinh \left (3 \, x + 2\right )}{2 \, {\left (\cosh \left (3 \, x + 2\right ) - \sinh \left (3 \, x + 2\right )\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(2+3*x)^2+2*sinh(2+3*x)^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 21, normalized size = 0.95 \[ \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (6 \, x + 4\right )} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(2+3*x)^2+2*sinh(2+3*x)^2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.52, size = 156, normalized size = 7.09 \[ \frac {\sqrt {6}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}-2 \sqrt {2}}\right )}{6 \sqrt {3}-6 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}-2 \sqrt {2}}\right )}{3 \left (2 \sqrt {3}-2 \sqrt {2}\right )}-\frac {\sqrt {6}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {3}+2 \sqrt {2}\right )}-\frac {2 \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {3}+2 \sqrt {2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(2+3*x)^2+2*sinh(2+3*x)^2),x)

[Out]

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

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maxima [A]  time = 0.43, size = 21, normalized size = 0.95 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (-6 \, x - 4\right )} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(2+3*x)^2+2*sinh(2+3*x)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.10, size = 21, normalized size = 0.95 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )}{4}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*sinh(3*x + 2)^2 + cosh(3*x + 2)^2),x)

[Out]

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

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sympy [B]  time = 7.89, size = 185, normalized size = 8.41 \[ \frac {2093258 \sqrt {5 - 2 \sqrt {6}} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {3 x}{2} + 1 \right )}}{\sqrt {5 - 2 \sqrt {6}}} \right )}}{1152360 \sqrt {6} + 2822694} + \frac {854569 \sqrt {6} \sqrt {5 - 2 \sqrt {6}} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {3 x}{2} + 1 \right )}}{\sqrt {5 - 2 \sqrt {6}}} \right )}}{1152360 \sqrt {6} + 2822694} - \frac {86329 \sqrt {6} \sqrt {2 \sqrt {6} + 5} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {3 x}{2} + 1 \right )}}{\sqrt {2 \sqrt {6} + 5}} \right )}}{1152360 \sqrt {6} + 2822694} - \frac {211462 \sqrt {2 \sqrt {6} + 5} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {3 x}{2} + 1 \right )}}{\sqrt {2 \sqrt {6} + 5}} \right )}}{1152360 \sqrt {6} + 2822694} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(2+3*x)**2+2*sinh(2+3*x)**2),x)

[Out]

2093258*sqrt(5 - 2*sqrt(6))*atan(tanh(3*x/2 + 1)/sqrt(5 - 2*sqrt(6)))/(1152360*sqrt(6) + 2822694) + 854569*sqr
t(6)*sqrt(5 - 2*sqrt(6))*atan(tanh(3*x/2 + 1)/sqrt(5 - 2*sqrt(6)))/(1152360*sqrt(6) + 2822694) - 86329*sqrt(6)
*sqrt(2*sqrt(6) + 5)*atan(tanh(3*x/2 + 1)/sqrt(2*sqrt(6) + 5))/(1152360*sqrt(6) + 2822694) - 211462*sqrt(2*sqr
t(6) + 5)*atan(tanh(3*x/2 + 1)/sqrt(2*sqrt(6) + 5))/(1152360*sqrt(6) + 2822694)

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