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

Optimal. Leaf size=3 \[ \tan ^{-1}(\tanh (x)) \]

[Out]

arctan(tanh(x))

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Rubi [A]  time = 0.02, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {203} \[ \tan ^{-1}(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^2 + Sinh[x]^2)^(-1),x]

[Out]

ArcTan[Tanh[x]]

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

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Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^2 + Sinh[x]^2)^(-1),x]

[Out]

ArcTan[Tanh[x]]

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fricas [B]  time = 0.43, size = 19, normalized size = 6.33 \[ -\arctan \left (-\frac {\cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(-(cosh(x) + sinh(x))/(cosh(x) - sinh(x)))

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giac [A]  time = 0.12, size = 5, normalized size = 1.67 \[ \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^(2*x))

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maple [B]  time = 0.23, size = 116, normalized size = 38.67 \[ \frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*2^(1/2)/(-2+2*2^(1/2))*arctan(2*tanh(1/2*x)/(-2+2*2^(1/2)))-2/(-2+2*2^(1/2))*arctan(2*tanh(1/2*x)/(-2+2*2^(1
/2)))-2*2^(1/2)/(2+2*2^(1/2))*arctan(2*tanh(1/2*x)/(2+2*2^(1/2)))-2/(2+2*2^(1/2))*arctan(2*tanh(1/2*x)/(2+2*2^
(1/2)))

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maxima [B]  time = 0.41, size = 35, normalized size = 11.67 \[ \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 5, normalized size = 1.67 \[ \mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2 + sinh(x)^2),x)

[Out]

atan(exp(2*x))

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sympy [B]  time = 7.66, size = 172, normalized size = 57.33 \[ \frac {47321 \sqrt {3 - 2 \sqrt {2}} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )}}{13860 \sqrt {2} + 19601} + \frac {33461 \sqrt {2} \sqrt {3 - 2 \sqrt {2}} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )}}{13860 \sqrt {2} + 19601} - \frac {5741 \sqrt {2} \sqrt {2 \sqrt {2} + 3} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )}}{13860 \sqrt {2} + 19601} - \frac {8119 \sqrt {2 \sqrt {2} + 3} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )}}{13860 \sqrt {2} + 19601} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

47321*sqrt(3 - 2*sqrt(2))*atan(tanh(x/2)/sqrt(3 - 2*sqrt(2)))/(13860*sqrt(2) + 19601) + 33461*sqrt(2)*sqrt(3 -
2*sqrt(2))*atan(tanh(x/2)/sqrt(3 - 2*sqrt(2)))/(13860*sqrt(2) + 19601) - 5741*sqrt(2)*sqrt(2*sqrt(2) + 3)*ata
n(tanh(x/2)/sqrt(2*sqrt(2) + 3))/(13860*sqrt(2) + 19601) - 8119*sqrt(2*sqrt(2) + 3)*atan(tanh(x/2)/sqrt(2*sqrt
(2) + 3))/(13860*sqrt(2) + 19601)

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