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

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

[Out]

-1/(2*b*d*(a + b*Sin[c + d*x])^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0263888, 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 = {2668, 32} \[ -\frac{1}{2 b d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*b*d*(a + b*Sin[c + d*x])^2)

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A]  time = 0.0244853, size = 22, normalized size = 1. \[ -\frac{1}{2 b d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*b*d*(a + b*Sin[c + d*x])^2)

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,bd \left ( a+b\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+b*sin(d*x+c))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.941628, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{2} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*sin(d*x + c) + a)^2*b*d)

________________________________________________________________________________________

Fricas [B]  time = 2.4696, size = 96, normalized size = 4.36 \begin{align*} \frac{1}{2 \,{\left (b^{3} d \cos \left (d x + c\right )^{2} - 2 \, a b^{2} d \sin \left (d x + c\right ) -{\left (a^{2} b + b^{3}\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.03525, size = 73, normalized size = 3.32 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{3}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{3} d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{3}} & \text{for}\: d = 0 \\- \frac{1}{2 a^{2} b d + 4 a b^{2} d \sin{\left (c + d x \right )} + 2 b^{3} d \sin ^{2}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.10792, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{2} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*sin(d*x + c) + a)^2*b*d)