3.253 \(\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{3 a d (a \sin (c+d x)+a)^3}-\frac{1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0520895, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{1}{3 a d (a \sin (c+d x)+a)^3}-\frac{1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^4,x]

[Out]

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

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{(a+x)^4}+\frac{1}{(a+x)^3}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{1}{3 a d (a+a \sin (c+d x))^3}-\frac{1}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.028238, size = 30, normalized size = 0.65 \[ -\frac{3 \sin (c+d x)+1}{6 a^4 d (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^4,x]

[Out]

-(1 + 3*Sin[c + d*x])/(6*a^4*d*(1 + Sin[c + d*x])^3)

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Maple [A]  time = 0.03, size = 33, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{4}} \left ( -{\frac{1}{2\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)/(a+a*sin(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.08795, size = 77, normalized size = 1.67 \begin{align*} -\frac{3 \, \sin \left (d x + c\right ) + 1}{6 \,{\left (a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/6*(3*sin(d*x + c) + 1)/((a^4*sin(d*x + c)^3 + 3*a^4*sin(d*x + c)^2 + 3*a^4*sin(d*x + c) + a^4)*d)

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Fricas [A]  time = 1.41514, size = 147, normalized size = 3.2 \begin{align*} \frac{3 \, \sin \left (d x + c\right ) + 1}{6 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(3*sin(d*x + c) + 1)/(3*a^4*d*cos(d*x + c)^2 - 4*a^4*d + (a^4*d*cos(d*x + c)^2 - 4*a^4*d)*sin(d*x + c))

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Sympy [A]  time = 3.86334, size = 129, normalized size = 2.8 \begin{align*} \begin{cases} - \frac{3 \sin{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} - \frac{1}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)/(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((-3*sin(c + d*x)/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*
a**4*d) - 1/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d), Ne(d,
0)), (x*sin(c)*cos(c)/(a*sin(c) + a)**4, True))

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Giac [A]  time = 1.1799, size = 38, normalized size = 0.83 \begin{align*} -\frac{3 \, \sin \left (d x + c\right ) + 1}{6 \, a^{4} d{\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(3*sin(d*x + c) + 1)/(a^4*d*(sin(d*x + c) + 1)^3)