3.792 \(\int \frac{a+b \cos (c+d x)}{(b+a \cos (c+d x))^2} \, dx\)

Optimal. Leaf size=22 \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+b)} \]

[Out]

Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))

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Rubi [A]  time = 0.0311741, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2754, 8} \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])/(b + a*Cos[c + d*x])^2,x]

[Out]

Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{a+b \cos (c+d x)}{(b+a \cos (c+d x))^2} \, dx &=\frac{\sin (c+d x)}{d (b+a \cos (c+d x))}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=\frac{\sin (c+d x)}{d (b+a \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0918501, size = 22, normalized size = 1. \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])/(b + a*Cos[c + d*x])^2,x]

[Out]

Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))

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Maple [B]  time = 0.052, size = 51, normalized size = 2.3 \begin{align*} -2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a- \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b-a-b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))/(b+cos(d*x+c)*a)^2,x)

[Out]

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

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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*cos(d*x+c))/(b+a*cos(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36325, size = 53, normalized size = 2.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))/(b+a*cos(d*x+c))^2,x, algorithm="fricas")

[Out]

sin(d*x + c)/(a*d*cos(d*x + c) + b*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))/(b+a*cos(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.2601, size = 68, normalized size = 3.09 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a - b\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))/(b+a*cos(d*x+c))^2,x, algorithm="giac")

[Out]

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