3.81 \(\int \sec (x) \, dx\)

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

[Out]

arctanh(sin(x))

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Rubi [A]  time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3770} \[ \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x],x]

[Out]

ArcTanh[Sin[x]]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \sec (x) \, dx &=\tanh ^{-1}(\sin (x))\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 33, normalized size = 11.00 \[ \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x],x]

[Out]

-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]

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fricas [B]  time = 0.43, size = 17, normalized size = 5.67 \[ \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x),x, algorithm="fricas")

[Out]

1/2*log(sin(x) + 1) - 1/2*log(-sin(x) + 1)

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giac [B]  time = 1.13, size = 17, normalized size = 5.67 \[ \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x),x, algorithm="giac")

[Out]

1/2*log(sin(x) + 1) - 1/2*log(-sin(x) + 1)

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maple [A]  time = 0.02, size = 7, normalized size = 2.33 \[ \ln \left (\sec \relax (x )+\tan \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x),x)

[Out]

ln(sec(x)+tan(x))

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maxima [B]  time = 0.42, size = 15, normalized size = 5.00 \[ \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\sin \relax (x) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x),x, algorithm="maxima")

[Out]

1/2*log(sin(x) + 1) - 1/2*log(sin(x) - 1)

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mupad [B]  time = 0.13, size = 11, normalized size = 3.67 \[ \ln \left (\frac {1}{\cos \relax (x)}\right )+\ln \left (\sin \relax (x)+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x),x)

[Out]

log(1/cos(x)) + log(sin(x) + 1)

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sympy [B]  time = 0.11, size = 15, normalized size = 5.00 \[ - \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x),x)

[Out]

-log(sin(x) - 1)/2 + log(sin(x) + 1)/2

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