### 3.80 $$\int \sec (x) (1-\sin (x)) \, dx$$

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

[Out]

Log[1 + Sin[x]]

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Rubi [A]  time = 0.013264, antiderivative size = 5, normalized size of antiderivative = 1., 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 = {2667, 31} $\log (\sin (x)+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sec[x]*(1 - Sin[x]),x]

[Out]

Log[1 + Sin[x]]

Rule 2667

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])

Rule 31

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

Rubi steps

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

Mathematica [B]  time = 0.0070203, size = 36, normalized size = 7.2 $\log (\cos (x))-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )$

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 = 0.021, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( 1+\sin \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1+sin(x))

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Maxima [A]  time = 0.935342, size = 7, normalized size = 1.4 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}

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 = 2.07607, size = 23, normalized size = 4.6 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}

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]  time = 0.317999, size = 19, normalized size = 3.8 \begin{align*} 2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} - \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} \end{align*}

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 = 1.05386, size = 7, normalized size = 1.4 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}

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)