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

Optimal. Leaf size=15 \[ \frac{a \sin (c+d x)}{d}+b x \]

[Out]

b*x + (a*Sin[c + d*x])/d

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Rubi [A]  time = 0.0231894, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3787, 2637, 8} \[ \frac{a \sin (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x + (a*Sin[c + d*x])/d

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0092314, size = 26, normalized size = 1.73 \[ \frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d

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Maple [A]  time = 0.034, size = 21, normalized size = 1.4 \begin{align*}{\frac{\sin \left ( dx+c \right ) a+ \left ( dx+c \right ) b}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(sin(d*x+c)*a+(d*x+c)*b)

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Maxima [A]  time = 1.16739, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} b + a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*x + c)*b + a*sin(d*x + c))/d

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Fricas [A]  time = 1.7471, size = 38, normalized size = 2.53 \begin{align*} \frac{b d x + a \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d*x + a*sin(d*x + c))/d

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Sympy [A]  time = 2.44176, size = 15, normalized size = 1. \begin{align*} a \left (\begin{cases} \sin{\left (c \right )} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{d} & \text{otherwise} \end{cases}\right ) + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Piecewise((sin(c), Eq(d, 0)), (sin(c + d*x)/d, True)) + b*x

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Giac [B]  time = 1.23696, size = 53, normalized size = 3.53 \begin{align*} \frac{{\left (d x + c\right )} b + \frac{2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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