3.2.12 \(\int \sec (a+b x) \, dx\) [112]

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

[Out]

arctanh(sin(b*x+a))/b

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3855} \begin {gather*} \frac {\tanh ^{-1}(\sin (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b

Rule 3855

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 (a+b x) \, dx &=\frac {\tanh ^{-1}(\sin (a+b x))}{b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(\sin (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b

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Maple [A]
time = 0.03, size = 19, normalized size = 1.73

method	result	size
derivativedivides	\(\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\)	\(19\)
default	\(\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b}\)	\(19\)
norman	\(\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}\)	\(35\)
risch	\(\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{b}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).
time = 6.20, size = 26, normalized size = 2.36 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right )}{2 \, b} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
time = 1.07, size = 28, normalized size = 2.55 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (8) = 16\).
time = 0.31, size = 34, normalized size = 3.09 \begin {gather*} \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} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
time = 0.86, size = 28, normalized size = 2.55 \begin {gather*} \frac {\log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(a + b*x),x)

[Out]

atanh(sin(a + b*x))/b

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