3.499 \(\int \frac{1}{5+3 \tan (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{3 \log (3 \sin (c+d x)+5 \cos (c+d x))}{34 d}+\frac{5 x}{34} \]

[Out]

(5*x)/34 + (3*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(34*d)

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Rubi [A]  time = 0.0374995, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3484, 3530} \[ \frac{3 \log (3 \sin (c+d x)+5 \cos (c+d x))}{34 d}+\frac{5 x}{34} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*Tan[c + d*x])^(-1),x]

[Out]

(5*x)/34 + (3*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(34*d)

Rule 3484

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2),
Int[(b - a*Tan[c + d*x])/(a + b*Tan[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 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{1}{5+3 \tan (c+d x)} \, dx &=\frac{5 x}{34}+\frac{3}{34} \int \frac{3-5 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx\\ &=\frac{5 x}{34}+\frac{3 \log (5 \cos (c+d x)+3 \sin (c+d x))}{34 d}\\ \end{align*}

Mathematica [C]  time = 0.0366271, size = 65, normalized size = 2.1 \[ -\frac{\left (\frac{3}{68}+\frac{5 i}{68}\right ) \log (-\tan (c+d x)+i)}{d}-\frac{\left (\frac{3}{68}-\frac{5 i}{68}\right ) \log (\tan (c+d x)+i)}{d}+\frac{3 \log (3 \tan (c+d x)+5)}{34 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*Tan[c + d*x])^(-1),x]

[Out]

((-3/68 - (5*I)/68)*Log[I - Tan[c + d*x]])/d - ((3/68 - (5*I)/68)*Log[I + Tan[c + d*x]])/d + (3*Log[5 + 3*Tan[
c + d*x]])/(34*d)

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Maple [A]  time = 0.015, size = 46, normalized size = 1.5 \begin{align*} -{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{68\,d}}+{\frac{5\,\arctan \left ( \tan \left ( dx+c \right ) \right ) }{34\,d}}+{\frac{3\,\ln \left ( 5+3\,\tan \left ( dx+c \right ) \right ) }{34\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*tan(d*x+c)),x)

[Out]

-3/68/d*ln(1+tan(d*x+c)^2)+5/34/d*arctan(tan(d*x+c))+3/34/d*ln(5+3*tan(d*x+c))

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Maxima [A]  time = 1.55479, size = 53, normalized size = 1.71 \begin{align*} \frac{10 \, d x + 10 \, c - 3 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, \log \left (3 \, \tan \left (d x + c\right ) + 5\right )}{68 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/68*(10*d*x + 10*c - 3*log(tan(d*x + c)^2 + 1) + 6*log(3*tan(d*x + c) + 5))/d

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Fricas [A]  time = 1.67566, size = 120, normalized size = 3.87 \begin{align*} \frac{10 \, d x + 3 \, \log \left (\frac{9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25}{\tan \left (d x + c\right )^{2} + 1}\right )}{68 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/68*(10*d*x + 3*log((9*tan(d*x + c)^2 + 30*tan(d*x + c) + 25)/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 0.564691, size = 46, normalized size = 1.48 \begin{align*} \begin{cases} \frac{5 x}{34} + \frac{3 \log{\left (\tan{\left (c + d x \right )} + \frac{5}{3} \right )}}{34 d} - \frac{3 \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{68 d} & \text{for}\: d \neq 0 \\\frac{x}{3 \tan{\left (c \right )} + 5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*tan(d*x+c)),x)

[Out]

Piecewise((5*x/34 + 3*log(tan(c + d*x) + 5/3)/(34*d) - 3*log(tan(c + d*x)**2 + 1)/(68*d), Ne(d, 0)), (x/(3*tan
(c) + 5), True))

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Giac [A]  time = 1.29952, size = 54, normalized size = 1.74 \begin{align*} \frac{10 \, d x + 10 \, c - 3 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, \log \left ({\left | 3 \, \tan \left (d x + c\right ) + 5 \right |}\right )}{68 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*tan(d*x+c)),x, algorithm="giac")

[Out]

1/68*(10*d*x + 10*c - 3*log(tan(d*x + c)^2 + 1) + 6*log(abs(3*tan(d*x + c) + 5)))/d