3.36 \(\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx\)

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

[Out]

a*x - (b*Log[Cos[c + d*x]])/d

________________________________________________________________________________________

Rubi [A]  time = 0.0257225, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3086, 3475} \[ a x-\frac{b \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*x - (b*Log[Cos[c + d*x]])/d

Rule 3086

Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

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 \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\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 = 0.0138661, size = 17, normalized size = 1. \[ a x-\frac{b \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*x - (b*Log[Cos[c + d*x]])/d

________________________________________________________________________________________

Maple [A]  time = 0.065, size = 24, normalized size = 1.4 \begin{align*} ax-{\frac{b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{ac}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.23574, size = 41, normalized size = 2.41 \begin{align*} \frac{2 \,{\left (d x + c\right )} a - b \log \left (-\sin \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.489085, size = 46, normalized size = 2.71 \begin{align*} \frac{a d x - b \log \left (-\cos \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*x - b*log(-cos(d*x + c)))/d

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cos{\left (c + d x \right )} + b \sin{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*cos(c + d*x) + b*sin(c + d*x))*sec(c + d*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.12282, size = 36, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (d x + c\right )} a + b \log \left (\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(sec(d*x+c)*(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")

[Out]

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