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

Optimal. Leaf size=61 \[ -\frac{7 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}-\frac{x}{a^3}+\frac{2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]

[Out]

-(x/a^3) - (7*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])) + (2*Cos[c + d*x])/(3*a*d*(a + a*Sin[c + d*x])^2)

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Rubi [A]  time = 0.108683, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2857, 2735, 2648} \[ -\frac{7 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}-\frac{x}{a^3}+\frac{2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a^3) - (7*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])) + (2*Cos[c + d*x])/(3*a*d*(a + a*Sin[c + d*x])^2)

Rule 2857

Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_
)]), x_Symbol] :> Simp[(2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(2*m + 3)), x] + Dist[
1/(b^3*(2*m + 3)), Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d*(2*m + 3)*Sin[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{2 \cos (c+d x)}{3 a d (a+a \sin (c+d x))^2}-\frac{\int \frac{-4 a+3 a \sin (c+d x)}{a+a \sin (c+d x)} \, dx}{3 a^3}\\ &=-\frac{x}{a^3}+\frac{2 \cos (c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac{7 \int \frac{1}{a+a \sin (c+d x)} \, dx}{3 a^2}\\ &=-\frac{x}{a^3}+\frac{2 \cos (c+d x)}{3 a d (a+a \sin (c+d x))^2}-\frac{7 \cos (c+d x)}{3 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.400779, size = 145, normalized size = 2.38 \[ -\frac{180 d x \sin \left (c+\frac{d x}{2}\right )+60 d x \sin \left (c+\frac{3 d x}{2}\right )+3 \sin \left (2 c+\frac{3 d x}{2}\right )-351 \cos \left (c+\frac{d x}{2}\right )+277 \cos \left (c+\frac{3 d x}{2}\right )-60 d x \cos \left (2 c+\frac{3 d x}{2}\right )-471 \sin \left (\frac{d x}{2}\right )+180 d x \cos \left (\frac{d x}{2}\right )}{120 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(180*d*x*Cos[(d*x)/2] - 351*Cos[c + (d*x)/2] + 277*Cos[c + (3*d*x)/2] - 60*d*x*Cos[2*c + (3*d*x)/2] - 471*Sin
[(d*x)/2] + 180*d*x*Sin[c + (d*x)/2] + 60*d*x*Sin[c + (3*d*x)/2] + 3*Sin[2*c + (3*d*x)/2])/(120*a^3*d*(Cos[c/2
] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)

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Maple [A]  time = 0.101, size = 83, normalized size = 1.4 \begin{align*} -2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}+{\frac{8}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-4\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-2\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/a^3*arctan(tan(1/2*d*x+1/2*c))+8/3/d/a^3/(tan(1/2*d*x+1/2*c)+1)^3-4/d/a^3/(tan(1/2*d*x+1/2*c)+1)^2-2/d/a^
3/(tan(1/2*d*x+1/2*c)+1)

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Maxima [B]  time = 1.72732, size = 192, normalized size = 3.15 \begin{align*} -\frac{2 \,{\left (\frac{\frac{12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5}{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}}} + \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*((12*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 5)/(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) +
3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

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Fricas [B]  time = 1.61444, size = 308, normalized size = 5.05 \begin{align*} -\frac{{\left (3 \, d x - 7\right )} \cos \left (d x + c\right )^{2} - 6 \, d x -{\left (3 \, d x + 5\right )} \cos \left (d x + c\right ) -{\left (6 \, d x +{\left (3 \, d x + 7\right )} \cos \left (d x + c\right ) + 2\right )} \sin \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((3*d*x - 7)*cos(d*x + c)^2 - 6*d*x - (3*d*x + 5)*cos(d*x + c) - (6*d*x + (3*d*x + 7)*cos(d*x + c) + 2)*s
in(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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Sympy [A]  time = 35.1207, size = 602, normalized size = 9.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-39*d*x*tan(c/2 + d*x/2)**3/(39*a**3*d*tan(c/2 + d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/2)**2 + 117*a
**3*d*tan(c/2 + d*x/2) + 39*a**3*d) - 117*d*x*tan(c/2 + d*x/2)**2/(39*a**3*d*tan(c/2 + d*x/2)**3 + 117*a**3*d*
tan(c/2 + d*x/2)**2 + 117*a**3*d*tan(c/2 + d*x/2) + 39*a**3*d) - 117*d*x*tan(c/2 + d*x/2)/(39*a**3*d*tan(c/2 +
d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/2)**2 + 117*a**3*d*tan(c/2 + d*x/2) + 39*a**3*d) - 39*d*x/(39*a**3*d*tan
(c/2 + d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/2)**2 + 117*a**3*d*tan(c/2 + d*x/2) + 39*a**3*d) + 36*tan(c/2 + d*
x/2)**3/(39*a**3*d*tan(c/2 + d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/2)**2 + 117*a**3*d*tan(c/2 + d*x/2) + 39*a**
3*d) + 30*tan(c/2 + d*x/2)**2/(39*a**3*d*tan(c/2 + d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/2)**2 + 117*a**3*d*tan
(c/2 + d*x/2) + 39*a**3*d) - 204*tan(c/2 + d*x/2)/(39*a**3*d*tan(c/2 + d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/2)
**2 + 117*a**3*d*tan(c/2 + d*x/2) + 39*a**3*d) - 94/(39*a**3*d*tan(c/2 + d*x/2)**3 + 117*a**3*d*tan(c/2 + d*x/
2)**2 + 117*a**3*d*tan(c/2 + d*x/2) + 39*a**3*d), Ne(d, 0)), (x*sin(c)*cos(c)**2/(a*sin(c) + a)**3, True))

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Giac [A]  time = 1.3235, size = 81, normalized size = 1.33 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*(d*x + c)/a^3 + 2*(3*tan(1/2*d*x + 1/2*c)^2 + 12*tan(1/2*d*x + 1/2*c) + 5)/(a^3*(tan(1/2*d*x + 1/2*c)
+ 1)^3))/d