3.112 \(\int \sec (a+b x) \, dx\)

Optimal. Leaf size=11 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]

[Out]

ArcTanh[Sin[a + b*x]]/b

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Rubi [A]  time = 0.00900368, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sec[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b

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Rubi in Sympy [A]  time = 0.611393, size = 8, normalized size = 0.73 \[ \frac{\operatorname{atanh}{\left (\sin{\left (a + b x \right )} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/cos(b*x+a),x)

[Out]

atanh(sin(a + b*x))/b

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Mathematica [B]  time = 0.017952, size = 68, normalized size = 6.18 \[ \frac{\log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )+\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )-\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[a + b*x],x]

[Out]

-(Log[Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2]]/b) + Log[Cos[a/2 + (b*x)/2] + Sin
[a/2 + (b*x)/2]]/b

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Maple [A]  time = 0.003, size = 19, normalized size = 1.7 \[{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/cos(b*x+a),x)

[Out]

1/b*ln(sec(b*x+a)+tan(b*x+a))

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Maxima [A]  time = 1.37825, size = 35, normalized size = 3.18 \[ \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/cos(b*x + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.232269, size = 38, normalized size = 3.45 \[ \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/cos(b*x + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.03065, size = 34, normalized size = 3.09 \[ \begin{cases} - \frac{\log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} - 1 \right )}}{b} + \frac{\log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} + 1 \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\cos{\left (a \right )}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/cos(b*x+a),x)

[Out]

Piecewise((-log(tan(a/2 + b*x/2) - 1)/b + log(tan(a/2 + b*x/2) + 1)/b, Ne(b, 0))
, (x/cos(a), True))

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GIAC/XCAS [A]  time = 0.216057, size = 38, normalized size = 3.45 \[ \frac{{\rm ln}\left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) -{\rm ln}\left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/cos(b*x + a),x, algorithm="giac")

[Out]

1/2*(ln(abs(sin(b*x + a) + 1)) - ln(abs(sin(b*x + a) - 1)))/b