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(d*x+c)/a/d

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [B]  time = 0.15, 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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fricas [A]  time = 0.43, size = 17, normalized size = 0.89 \[ \frac {d x + \cos \left (d x + c\right )}{a d} \]

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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giac [A]  time = 0.67, size = 34, normalized size = 1.79 \[ \frac {\frac {d x + c}{a} + \frac {2}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a}}{d} \]

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

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maple [B]  time = 0.10, size = 43, normalized size = 2.26 \[ \frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]

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 = 0.74, size = 52, normalized size = 2.74 \[ \frac {2 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}\right )}}{d} \]

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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mupad [B]  time = 2.77, size = 29, normalized size = 1.53 \[ \frac {x}{a}+\frac {2}{a\,d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.77, size = 88, normalized size = 4.63 \[ \begin {cases} \frac {d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {d x}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {2}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{2}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]

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) + 2/(
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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