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(d*x+c)/d

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 8

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

Rule 2637

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

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]

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 = 0.01, 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

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 17, normalized size = 1.13 \[ \frac {b d x + a \sin \left (d x + c\right )}{d} \]

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

________________________________________________________________________________________

giac [B]  time = 0.16, size = 39, normalized size = 2.60 \[ \frac {{\left (d x + c\right )} b + \frac {2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]

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

________________________________________________________________________________________

maple [A]  time = 0.50, size = 21, normalized size = 1.40 \[ \frac {a \sin \left (d x +c \right )+\left (d x +c \right ) b}{d} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 20, normalized size = 1.33 \[ \frac {{\left (d x + c\right )} b + a \sin \left (d x + c\right )}{d} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.74, size = 17, normalized size = 1.13 \[ \frac {a\,\sin \left (c+d\,x\right )+b\,d\,x}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(c + d*x) + b*d*x)/d

________________________________________________________________________________________

sympy [A]  time = 1.86, size = 15, normalized size = 1.00 \[ a \left (\begin {cases} \sin {\relax (c )} & \text {for}\: d = 0 \\\frac {\sin {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) + b x \]

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

________________________________________________________________________________________