3.314 \(\int \frac{3+\tan (c+d x)}{2-\tan (c+d x)} \, dx\)

Optimal. Leaf size=25 \[ x-\frac{\log (2 \cos (c+d x)-\sin (c+d x))}{d} \]

[Out]

x - Log[2*Cos[c + d*x] - Sin[c + d*x]]/d

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Rubi [A]  time = 0.0483056, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3531, 3530} \[ x-\frac{\log (2 \cos (c+d x)-\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(3 + Tan[c + d*x])/(2 - Tan[c + d*x]),x]

[Out]

x - Log[2*Cos[c + d*x] - Sin[c + d*x]]/d

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{3+\tan (c+d x)}{2-\tan (c+d x)} \, dx &=x-\int \frac{-1-2 \tan (c+d x)}{2-\tan (c+d x)} \, dx\\ &=x-\frac{\log (2 \cos (c+d x)-\sin (c+d x))}{d}\\ \end{align*}

Mathematica [B]  time = 0.0416813, size = 62, normalized size = 2.48 \[ \frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\log \left ((2-\tan (c+d x))^2-4 (2-\tan (c+d x))+5\right )}{2 d}-\frac{\log (2-\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + Tan[c + d*x])/(2 - Tan[c + d*x]),x]

[Out]

ArcTan[Tan[c + d*x]]/d + Log[5 - 4*(2 - Tan[c + d*x]) + (2 - Tan[c + d*x])^2]/(2*d) - Log[2 - Tan[c + d*x]]/d

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Maple [A]  time = 0.024, size = 41, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{\ln \left ( -2+\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{dx+c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+tan(d*x+c))/(2-tan(d*x+c)),x)

[Out]

1/2/d*ln(1+tan(d*x+c)^2)-1/d*ln(-2+tan(d*x+c))+1/d*(d*x+c)

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Maxima [A]  time = 1.80106, size = 47, normalized size = 1.88 \begin{align*} \frac{2 \, d x + 2 \, c + \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, \log \left (\tan \left (d x + c\right ) - 2\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+tan(d*x+c))/(2-tan(d*x+c)),x, algorithm="maxima")

[Out]

1/2*(2*d*x + 2*c + log(tan(d*x + c)^2 + 1) - 2*log(tan(d*x + c) - 2))/d

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Fricas [A]  time = 1.64685, size = 109, normalized size = 4.36 \begin{align*} \frac{2 \, d x - \log \left (\frac{\tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right ) + 4}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+tan(d*x+c))/(2-tan(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(2*d*x - log((tan(d*x + c)^2 - 4*tan(d*x + c) + 4)/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 0.406301, size = 39, normalized size = 1.56 \begin{align*} \begin{cases} x - \frac{\log{\left (\tan{\left (c + d x \right )} - 2 \right )}}{d} + \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\\frac{x \left (\tan{\left (c \right )} + 3\right )}{2 - \tan{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+tan(d*x+c))/(2-tan(d*x+c)),x)

[Out]

Piecewise((x - log(tan(c + d*x) - 2)/d + log(tan(c + d*x)**2 + 1)/(2*d), Ne(d, 0)), (x*(tan(c) + 3)/(2 - tan(c
)), True))

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Giac [A]  time = 1.20173, size = 49, normalized size = 1.96 \begin{align*} \frac{2 \, d x + 2 \, c + \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (d x + c\right ) - 2 \right |}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+tan(d*x+c))/(2-tan(d*x+c)),x, algorithm="giac")

[Out]

1/2*(2*d*x + 2*c + log(tan(d*x + c)^2 + 1) - 2*log(abs(tan(d*x + c) - 2)))/d