3.283 \(\int \frac{\cos (c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{B \sin (c+d x)}{d} \]

[Out]

(B*Sin[c + d*x])/d

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Rubi [A]  time = 0.0087709, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {21, 2637} \[ \frac{B \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*Sin[c + d*x])/d

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 2637

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

Rubi steps

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

Mathematica [B]  time = 0.0073006, size = 23, normalized size = 2.09 \[ B \left (\frac{\sin (c) \cos (d x)}{d}+\frac{\cos (c) \sin (d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 12, normalized size = 1.1 \begin{align*}{\frac{B\sin \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*sin(d*x+c)/d

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40392, size = 24, normalized size = 2.18 \begin{align*} \frac{B \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+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

B*sin(d*x + c)/d

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Sympy [A]  time = 0.746644, size = 31, normalized size = 2.82 \begin{align*} \begin{cases} \frac{B \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (B a + B b \cos{\left (c \right )}\right ) \cos{\left (c \right )}}{a + b \cos{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((B*sin(c + d*x)/d, Ne(d, 0)), (x*(B*a + B*b*cos(c))*cos(c)/(a + b*cos(c)), True))

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Giac [A]  time = 1.39754, size = 15, normalized size = 1.36 \begin{align*} \frac{B \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+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

B*sin(d*x + c)/d