3.81 \(\int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=27 \[ -\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]

[Out]

-1/3*cos(d*x+c)^3/d/(a+a*sin(d*x+c))^3

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Rubi [A]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2671} \[ -\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]

[Out]

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

Rule 2671

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.04 \[ -\frac {\cos ^3(c+d x)}{3 a^3 d (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]

[Out]

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

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fricas [B]  time = 0.47, size = 95, normalized size = 3.52 \[ -\frac {\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/3*(cos(d*x + c)^2 + (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)/(a^3*d*cos(d*x + c)^2 - a^3*d*cos(d
*x + c) - 2*a^3*d - (a^3*d*cos(d*x + c) + 2*a^3*d)*sin(d*x + c))

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giac [A]  time = 0.55, size = 36, normalized size = 1.33 \[ -\frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{3 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-2/3*(3*tan(1/2*d*x + 1/2*c)^2 + 1)/(a^3*d*(tan(1/2*d*x + 1/2*c) + 1)^3)

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maple [B]  time = 0.22, size = 55, normalized size = 2.04 \[ \frac {-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2/(a+a*sin(d*x+c))^3,x)

[Out]

2/d/a^3*(-4/3/(tan(1/2*d*x+1/2*c)+1)^3+2/(tan(1/2*d*x+1/2*c)+1)^2-1/(tan(1/2*d*x+1/2*c)+1))

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maxima [B]  time = 0.43, size = 99, normalized size = 3.67 \[ -\frac {2 \, {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{3 \, {\left (a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-2/3*(3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/((a^3 + 3*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 3*a^3*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)*d)

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mupad [B]  time = 4.58, size = 53, normalized size = 1.96 \[ \frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-3\right )}{3\,a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^2/(a + a*sin(c + d*x))^3,x)

[Out]

(2*cos(c/2 + (d*x)/2)*(2*cos(c/2 + (d*x)/2)^2 - 3))/(3*a^3*d*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))^3)

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sympy [A]  time = 15.15, size = 153, normalized size = 5.67 \[ \begin {cases} - \frac {6 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {2}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{2}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2/(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((-6*tan(c/2 + d*x/2)**2/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/2)**2 + 9*a**3*d*tan(
c/2 + d*x/2) + 3*a**3*d) - 2/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/2)**2 + 9*a**3*d*tan(c/2 +
d*x/2) + 3*a**3*d), Ne(d, 0)), (x*cos(c)**2/(a*sin(c) + a)**3, True))

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