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

Optimal. Leaf size=3 \[ B x \]

[Out]

B*x

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Rubi [A]  time = 0.0011028, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {21, 8} \[ B x \]

Antiderivative was successfully verified.

[In]

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

[Out]

B*x

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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0002416, size = 3, normalized size = 1. \[ B x \]

Antiderivative was successfully verified.

[In]

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

[Out]

B*x

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Maple [A]  time = 0.025, size = 4, normalized size = 1.3 \begin{align*} Bx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

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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((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 = 1.1665, size = 7, normalized size = 2.33 \begin{align*} B x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

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Sympy [A]  time = 0.158853, size = 2, normalized size = 0.67 \begin{align*} B x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x

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Giac [C]  time = 1.32111, size = 14, normalized size = 4.67 \begin{align*} \frac{{\left (d x + c\right )} B}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)*B/d