3.290 \(\int \frac{\tan (c+d x)}{a+b \sec (c+d x)} \, dx\)

Optimal. Leaf size=35 \[ -\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d} \]

[Out]

-(Log[Cos[c + d*x]]/(a*d)) - Log[a + b*Sec[c + d*x]]/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0315418, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3885, 36, 29, 31} \[ -\frac{\log (a+b \sec (c+d x))}{a d}-\frac{\log (\cos (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]/(a + b*Sec[c + d*x]),x]

[Out]

-(Log[Cos[c + d*x]]/(a*d)) - Log[a + b*Sec[c + d*x]]/(a*d)

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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{\tan (c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,b \sec (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sec (c+d x)\right )}{a d}\\ &=-\frac{\log (\cos (c+d x))}{a d}-\frac{\log (a+b \sec (c+d x))}{a d}\\ \end{align*}

Mathematica [A]  time = 0.0368242, size = 19, normalized size = 0.54 \[ -\frac{\log (a \cos (c+d x)+b)}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]/(a + b*Sec[c + d*x]),x]

[Out]

-(Log[b + a*Cos[c + d*x]]/(a*d))

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 35, normalized size = 1. \begin{align*} -{\frac{\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{ad}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{ad}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(a+b*sec(d*x+c))/d/a+1/d/a*ln(sec(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 0.963872, size = 26, normalized size = 0.74 \begin{align*} -\frac{\log \left (a \cos \left (d x + c\right ) + b\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*cos(d*x + c) + b)/(a*d)

________________________________________________________________________________________

Fricas [A]  time = 0.927896, size = 43, normalized size = 1.23 \begin{align*} -\frac{\log \left (a \cos \left (d x + c\right ) + b\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*cos(d*x + c) + b)/(a*d)

________________________________________________________________________________________

Sympy [A]  time = 6.82994, size = 82, normalized size = 2.34 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \tan{\left (c \right )}}{\sec{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{x \tan{\left (c \right )}}{a + b \sec{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{1}{b d \sec{\left (c + d x \right )}} & \text{for}\: a = 0 \\\frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} & \text{for}\: b = 0 \\- \frac{\log{\left (\frac{a}{b} + \sec{\left (c + d x \right )} \right )}}{a d} + \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*tan(c)/sec(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (x*tan(c)/(a + b*sec(c)), Eq(d, 0)), (-1/(b*d
*sec(c + d*x)), Eq(a, 0)), (log(tan(c + d*x)**2 + 1)/(2*a*d), Eq(b, 0)), (-log(a/b + sec(c + d*x))/(a*d) + log
(tan(c + d*x)**2 + 1)/(2*a*d), True))

________________________________________________________________________________________

Giac [B]  time = 1.35047, size = 131, normalized size = 3.74 \begin{align*} -\frac{\frac{{\left (a - b\right )} \log \left ({\left | a + b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{2} - a b} - \frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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