3.1.80 \(\int \sec (x) (1-\sin (x)) \, dx\) [80]

Optimal. Leaf size=5 \[ \log (1+\sin (x)) \]

[Out]

ln(1+sin(x))

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Rubi [A]
time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2746, 31} \begin {gather*} \log (\sin (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]*(1 - Sin[x]),x]

[Out]

Log[1 + Sin[x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \sec (x) (1-\sin (x)) \, dx &=-\text {Subst}\left (\int \frac {1}{1-x} \, dx,x,-\sin (x)\right )\\ &=\log (1+\sin (x))\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(5)=10\).
time = 0.01, size = 36, normalized size = 7.20 \begin {gather*} \log (\cos (x))-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]*(1 - Sin[x]),x]

[Out]

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

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Maple [A]
time = 0.04, size = 6, normalized size = 1.20

method	result	size
derivativedivides	\(\ln \left (\sin \left (x \right )+1\right )\)	\(6\)
default	\(\ln \left (\sin \left (x \right )+1\right )\)	\(6\)
risch	\(-i x +2 \ln \left ({\mathrm e}^{i x}+i\right )\)	\(17\)
norman	\(2 \ln \left (1+\tan \left (\frac {x}{2}\right )\right )-\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-sin(x))/cos(x),x,method=_RETURNVERBOSE)

[Out]

ln(sin(x)+1)

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Maxima [A]
time = 3.09, size = 5, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x) + 1)

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Fricas [A]
time = 0.96, size = 5, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x) + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (5) = 10\).
time = 0.12, size = 19, normalized size = 3.80 \begin {gather*} 2 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} - \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sin(x))/cos(x),x)

[Out]

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

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Giac [A]
time = 1.13, size = 5, normalized size = 1.00 \begin {gather*} \log \left (\sin \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(x) + 1)

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Mupad [B]
time = 0.17, size = 5, normalized size = 1.00 \begin {gather*} \ln \left (\sin \left (x\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(sin(x) - 1)/cos(x),x)

[Out]

log(sin(x) + 1)

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