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

Optimal. Leaf size=47 \[ \frac{\cos (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{2 x}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0724305, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2857, 2638} \[ \frac{\cos (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{2 x}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [B]  time = 0.317242, size = 117, normalized size = 2.49 \[ \frac{12 d x \sin \left (c+\frac{d x}{2}\right )+3 \sin \left (2 c+\frac{3 d x}{2}\right )+2 \cos \left (c+\frac{d x}{2}\right )+3 \cos \left (c+\frac{3 d x}{2}\right )-28 \sin \left (\frac{d x}{2}\right )+12 d x \cos \left (\frac{d x}{2}\right )}{6 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*d*x*Cos[(d*x)/2] + 2*Cos[c + (d*x)/2] + 3*Cos[c + (3*d*x)/2] - 28*Sin[(d*x)/2] + 12*d*x*Sin[c + (d*x)/2] +
3*Sin[2*c + (3*d*x)/2])/(6*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 0.081, size = 64, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}+4\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}}+4\,{\frac{1}{d{a}^{2} \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))^2,x)

[Out]

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

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Maxima [B]  time = 1.70188, size = 188, normalized size = 4. \begin{align*} \frac{2 \,{\left (\frac{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 3}{a^{2} + \frac{a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.66963, size = 201, normalized size = 4.28 \begin{align*} \frac{2 \, d x +{\left (2 \, d x + 3\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} +{\left (2 \, d x + \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 2}{a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} 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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 14.5296, size = 479, normalized size = 10.19 \begin{align*} \begin{cases} \frac{2 d x \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} + \frac{2 d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} + \frac{2 d x \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} + \frac{2 d x}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} + \frac{4 \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} + \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} + \frac{6}{a^{2} d \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )} \cos ^{2}{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \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))**2,x)

[Out]

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

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Giac [A]  time = 1.25503, size = 105, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (\frac{d x + c}{a^{2}} + \frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )} a^{2}}\right )}}{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))^2,x, algorithm="giac")

[Out]

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