3.65 \(\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac{(a-a \sin (c+d x))^3}{3 a^5 d} \]

[Out]

-(a - a*Sin[c + d*x])^3/(3*a^5*d)

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Rubi [A]  time = 0.0431779, antiderivative size = 23, 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 = {2667, 32} \[ -\frac{(a-a \sin (c+d x))^3}{3 a^5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]

[Out]

-(a - a*Sin[c + d*x])^3/(3*a^5*d)

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac{(a-a \sin (c+d x))^3}{3 a^5 d}\\ \end{align*}

Mathematica [A]  time = 0.0561295, size = 34, normalized size = 1.48 \[ \frac{\sin (c+d x) \left (\sin ^2(c+d x)-3 \sin (c+d x)+3\right )}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]

[Out]

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

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Maple [A]  time = 0.062, size = 19, normalized size = 0.8 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}{3\,d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+a*sin(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 0.939798, size = 47, normalized size = 2.04 \begin{align*} \frac{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right )}{3 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.90076, size = 92, normalized size = 4. \begin{align*} \frac{3 \, \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right )}{3 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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Sympy [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(cos(d*x+c)**5/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.1311, size = 47, normalized size = 2.04 \begin{align*} \frac{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right )}{3 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

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