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

Optimal. Leaf size=22 \[ -\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

[Out]

-1/(2*a*d*(a + a*Sin[c + d*x])^2)

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Rubi [A]  time = 0.0246114, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 32} \[ -\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a*d*(a + a*Sin[c + d*x])^2)

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 (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{1}{2 a d (a+a \sin (c+d x))^2}\\ \end{align*}

Mathematica [A]  time = 0.0569452, size = 33, normalized size = 1.5 \[ -\frac{1}{2 a^3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

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Maple [A]  time = 0.014, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,da \left ( a+a\sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.94394, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a*sin(d*x + c) + a)^2*a*d)

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Fricas [A]  time = 1.83076, size = 82, normalized size = 3.73 \begin{align*} \frac{1}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.66904, size = 51, normalized size = 2.32 \begin{align*} \begin{cases} - \frac{1}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(2*a**3*d*sin(c + d*x)**2 + 4*a**3*d*sin(c + d*x) + 2*a**3*d), Ne(d, 0)), (x*cos(c)/(a*sin(c) +
a)**3, True))

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Giac [A]  time = 1.19321, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a*sin(d*x + c) + a)^2*a*d)