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

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

[Out]

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

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Rubi [A]  time = 0.0367905, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3884, 3475, 2606, 8} \[ \frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3884

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(e*
Cot[c + d*x])^m, x], x] + Dist[b, Int[(e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]

Rule 3475

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

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0139561, size = 25, normalized size = 1. \[ \frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.016, size = 25, normalized size = 1. \begin{align*}{\frac{a\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}+{\frac{b\sec \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992142, size = 35, normalized size = 1.4 \begin{align*} -\frac{a \log \left (\cos \left (d x + c\right )\right ) - \frac{b}{\cos \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*log(cos(d*x + c)) - b/cos(d*x + c))/d

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Fricas [A]  time = 0.770841, size = 80, normalized size = 3.2 \begin{align*} -\frac{a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - b}{d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*cos(d*x + c)*log(-cos(d*x + c)) - b)/(d*cos(d*x + c))

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Sympy [A]  time = 0.98261, size = 37, normalized size = 1.48 \begin{align*} \begin{cases} \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \sec{\left (c \right )}\right ) \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20217, size = 144, normalized size = 5.76 \begin{align*} \frac{a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{a + 2 \, b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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