3.417 \(\int (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=17 \[ a x-\frac{b \log (\cos (c+d x))}{d} \]

[Out]

a*x - (b*Log[Cos[c + d*x]])/d

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Rubi [A]  time = 0.0071293, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3475} \[ a x-\frac{b \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Tan[c + d*x],x]

[Out]

a*x - (b*Log[Cos[c + d*x]])/d

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (a+b \tan (c+d x)) \, dx &=a x+b \int \tan (c+d x) \, dx\\ &=a x-\frac{b \log (\cos (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.0046165, size = 17, normalized size = 1. \[ a x-\frac{b \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Tan[c + d*x],x]

[Out]

a*x - (b*Log[Cos[c + d*x]])/d

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Maple [A]  time = 0.002, size = 22, normalized size = 1.3 \begin{align*} ax+{\frac{b\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*tan(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.05051, size = 22, normalized size = 1.29 \begin{align*} a x + \frac{b \log \left (\sec \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c),x, algorithm="maxima")

[Out]

a*x + b*log(sec(d*x + c))/d

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Fricas [A]  time = 1.72243, size = 66, normalized size = 3.88 \begin{align*} \frac{2 \, a d x - b \log \left (\frac{1}{\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(a+b*tan(d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.130422, size = 24, normalized size = 1.41 \begin{align*} a x + b \left (\begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\x \tan{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.25719, size = 24, normalized size = 1.41 \begin{align*} a x - \frac{b \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c),x, algorithm="giac")

[Out]

a*x - b*log(abs(cos(d*x + c)))/d