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

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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*}

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Mathematica [A]  time = 0.05, 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)

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fricas [A]  time = 0.97, size = 36, normalized size = 1.38 \[ \frac {2 \, \sqrt {a \cos \left (d x + c\right ) + a} B \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]

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)

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giac [A]  time = 0.96, size = 35, normalized size = 1.35 \[ \frac {2 \, \sqrt {2} B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]

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)

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maple [A]  time = 0.23, size = 43, normalized size = 1.65 \[ \frac {2 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}}{\sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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mupad [B]  time = 0.74, size = 37, normalized size = 1.42 \[ \frac {2\,B\,\sin \left (c+d\,x\right )\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}}{a\,d\,\left (\cos \left (c+d\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ B \left (\int \frac {\cos {\left (c + d x \right )}}{\sqrt {a \cos {\left (c + d x \right )} + a}}\, dx + \int \frac {1}{\sqrt {a \cos {\left (c + d x \right )} + a}}\, dx\right ) \]

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))

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