3.1.5 \(\int (a+a \sec (c+d x)) \, dx\) [5]

Optimal. Leaf size=16 \[ a x+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

a*x+a*arctanh(sin(d*x+c))/d

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3855} \begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+a x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + a*Sec[c + d*x],x]

[Out]

a*x + (a*ArcTanh[Sin[c + d*x]])/d

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} a x+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + a*Sec[c + d*x],x]

[Out]

a*x + (a*ArcTanh[Sin[c + d*x]])/d

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Maple [A]
time = 0.03, size = 24, normalized size = 1.50

method	result	size
default	\(a x +\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\)	\(24\)
derivativedivides	\(\frac {\left (d x +c \right ) a +a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\)	\(29\)
norman	\(a x +\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\)	\(40\)
risch	\(a x -\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+a*sec(d*x+c),x,method=_RETURNVERBOSE)

[Out]

a*x+a/d*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]
time = 0.29, size = 23, normalized size = 1.44 \begin {gather*} a x + \frac {a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+a*sec(d*x+c),x, algorithm="maxima")

[Out]

a*x + a*log(sec(d*x + c) + tan(d*x + c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
time = 2.81, size = 36, normalized size = 2.25 \begin {gather*} \frac {2 \, a d x + a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+a*sec(d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.09, size = 41, normalized size = 2.56 \begin {gather*} a x + a \left (\begin {cases} \frac {\log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (\tan {\left (c \right )} \sec {\left (c \right )} + \sec ^{2}{\left (c \right )}\right )}{\tan {\left (c \right )} + \sec {\left (c \right )}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+a*sec(d*x+c),x)

[Out]

a*x + a*Piecewise((log(tan(c + d*x) + sec(c + d*x))/d, Ne(d, 0)), (x*(tan(c)*sec(c) + sec(c)**2)/(tan(c) + sec
(c)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (16) = 32\).
time = 0.45, size = 49, normalized size = 3.06 \begin {gather*} a x + \frac {a {\left (\log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+a*sec(d*x+c),x, algorithm="giac")

[Out]

a*x + 1/4*a*(log(abs(1/sin(d*x + c) + sin(d*x + c) + 2)) - log(abs(1/sin(d*x + c) + sin(d*x + c) - 2)))/d

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Mupad [B]
time = 0.61, size = 20, normalized size = 1.25 \begin {gather*} a\,x+\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + a/cos(c + d*x),x)

[Out]

a*x + (2*a*atanh(tan(c/2 + (d*x)/2)))/d

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