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

Optimal. Leaf size=19 \[ \frac{\cos (c+d x)}{a d}+\frac{x}{a} \]

[Out]

x/a + Cos[c + d*x]/(a*d)

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Rubi [A]  time = 0.0421265, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2682, 8} \[ \frac{\cos (c+d x)}{a d}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a + Cos[c + d*x]/(a*d)

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\cos (c+d x)}{a d}+\frac{\int 1 \, dx}{a}\\ &=\frac{x}{a}+\frac{\cos (c+d x)}{a d}\\ \end{align*}

Mathematica [B]  time = 0.138564, size = 97, normalized size = 5.11 \[ -\frac{\left (2 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )+(\sin (c+d x)-1) \sqrt{\sin (c+d x)+1}\right ) \cos ^3(c+d x)}{a d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Cos[c + d*x]^3*(2*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + (-1 + Sin[c + d*x])*Sqrt[
1 + Sin[c + d*x]]))/(a*d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2)))

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Maple [B]  time = 0.048, size = 43, normalized size = 2.3 \begin{align*} 2\,{\frac{1}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.59223, size = 70, normalized size = 3.68 \begin{align*} \frac{2 \,{\left (\frac{\arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{1}{a + \frac{a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59846, size = 38, normalized size = 2. \begin{align*} \frac{d x + \cos \left (d x + c\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + cos(d*x + c))/(a*d)

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Sympy [A]  time = 3.26248, size = 119, normalized size = 6.26 \begin{align*} \begin{cases} \frac{d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{d x}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{\tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{1}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x \cos ^{2}{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d*x*tan(c/2 + d*x/2)**2/(a*d*tan(c/2 + d*x/2)**2 + a*d) + d*x/(a*d*tan(c/2 + d*x/2)**2 + a*d) - tan
(c/2 + d*x/2)**2/(a*d*tan(c/2 + d*x/2)**2 + a*d) + 1/(a*d*tan(c/2 + d*x/2)**2 + a*d), Ne(d, 0)), (x*cos(c)**2/
(a*sin(c) + a), True))

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Giac [A]  time = 1.11696, size = 46, normalized size = 2.42 \begin{align*} \frac{\frac{d x + c}{a} + \frac{2}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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