3.690 \(\int \frac{\sec ^2(x)}{a+b \tan (x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{\log (a+b \tan (x))}{b} \]

[Out]

Log[a + b*Tan[x]]/b

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Rubi [A]  time = 0.0341299, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3506, 31} \[ \frac{\log (a+b \tan (x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2/(a + b*Tan[x]),x]

[Out]

Log[a + b*Tan[x]]/b

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/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 \frac{\sec ^2(x)}{a+b \tan (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (x)\right )}{b}\\ &=\frac{\log (a+b \tan (x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0652282, size = 20, normalized size = 1.82 \[ \frac{\log (a \cos (x)+b \sin (x))-\log (\cos (x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2/(a + b*Tan[x]),x]

[Out]

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

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Maple [A]  time = 0.027, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2/(a+b*tan(x)),x)

[Out]

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

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Maxima [A]  time = 0.953026, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \tan \left (x\right ) + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(a+b*tan(x)),x, algorithm="maxima")

[Out]

log(b*tan(x) + a)/b

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Fricas [B]  time = 2.25314, size = 107, normalized size = 9.73 \begin{align*} \frac{\log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - \log \left (\cos \left (x\right )^{2}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(a+b*tan(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (x \right )}}{a + b \tan{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2/(a+b*tan(x)),x)

[Out]

Integral(sec(x)**2/(a + b*tan(x)), x)

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Giac [A]  time = 1.08348, size = 16, normalized size = 1.45 \begin{align*} \frac{\log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(a+b*tan(x)),x, algorithm="giac")

[Out]

log(abs(b*tan(x) + a))/b