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

Optimal. Leaf size=26 \[ \frac{2 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0155015, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {21, 2646} \[ \frac{2 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*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 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{B+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx &=\frac{B \int \sqrt{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac{2 B \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0427348, size = 33, normalized size = 1.27 \[ \frac{2 B \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/(a*d)

________________________________________________________________________________________

Maple [A]  time = 0.904, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{B\cos \left ( 1/2\,dx+c/2 \right ) \sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B+B*cos(d*x+c))/(a+cos(d*x+c)*a)^(1/2),x)

[Out]

2*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)*2^(1/2)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

________________________________________________________________________________________

Maxima [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((B+B*cos(d*x+c))/(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 1.3451, size = 92, normalized size = 3.54 \begin{align*} \frac{2 \, \sqrt{a \cos \left (d x + c\right ) + a} B \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B+B*cos(d*x+c))/(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(a*cos(d*x + c) + a)*B*sin(d*x + c)/(a*d*cos(d*x + c) + a*d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} B \left (\int \frac{\cos{\left (c + d x \right )}}{\sqrt{a \cos{\left (c + d x \right )} + a}}\, dx + \int \frac{1}{\sqrt{a \cos{\left (c + d x \right )} + a}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B+B*cos(d*x+c))/(a+a*cos(d*x+c))**(1/2),x)

[Out]

B*(Integral(cos(c + d*x)/sqrt(a*cos(c + d*x) + a), x) + Integral(1/sqrt(a*cos(c + d*x) + a), x))

________________________________________________________________________________________

Giac [A]  time = 2.44693, size = 47, normalized size = 1.81 \begin{align*} \frac{2 \, \sqrt{2} B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B+B*cos(d*x+c))/(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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