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

Optimal. Leaf size=45 \[ -\frac{\cos (c+d x)}{a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.072349, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2839, 2638, 2635, 8} \[ -\frac{\cos (c+d x)}{a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2638

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [B]  time = 0.561115, size = 161, normalized size = 3.58 \[ \frac{-4 d x \sin \left (\frac{c}{2}\right )+4 \sin \left (\frac{c}{2}+d x\right )-4 \sin \left (\frac{3 c}{2}+d x\right )+\sin \left (\frac{3 c}{2}+2 d x\right )+\sin \left (\frac{5 c}{2}+2 d x\right )+2 \cos \left (\frac{c}{2}\right ) (c-2 d x)-4 \cos \left (\frac{c}{2}+d x\right )-4 \cos \left (\frac{3 c}{2}+d x\right )+\cos \left (\frac{3 c}{2}+2 d x\right )-\cos \left (\frac{5 c}{2}+2 d x\right )+2 c \sin \left (\frac{c}{2}\right )-4 \sin \left (\frac{c}{2}\right )}{8 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c - 2*d*x)*Cos[c/2] - 4*Cos[c/2 + d*x] - 4*Cos[(3*c)/2 + d*x] + Cos[(3*c)/2 + 2*d*x] - Cos[(5*c)/2 + 2*d*x
] - 4*Sin[c/2] + 2*c*Sin[c/2] - 4*d*x*Sin[c/2] + 4*Sin[c/2 + d*x] - 4*Sin[(3*c)/2 + d*x] + Sin[(3*c)/2 + 2*d*x
] + Sin[(5*c)/2 + 2*d*x])/(8*a*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.06, size = 142, normalized size = 3.2 \begin{align*} -{\frac{1}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{1}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{1}{da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \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)),x)

[Out]

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

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Maxima [B]  time = 1.59499, size = 180, normalized size = 4. \begin{align*} \frac{\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}} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 2}{a + \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac{\arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57502, size = 85, normalized size = 1.89 \begin{align*} -\frac{d x - \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )}{2 \, a 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)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.31351, size = 366, normalized size = 8.13 \begin{align*} \begin{cases} - \frac{d x \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} - \frac{2 d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} - \frac{d x}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} + \frac{2 \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} - \frac{2 \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} + \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} - \frac{2}{2 a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 4 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )} \cos ^{2}{\left (c \right )}}{a \sin{\left (c \right )} + a} & \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)),x)

[Out]

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

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Giac [A]  time = 1.28341, size = 97, normalized size = 2.16 \begin{align*} -\frac{\frac{d x + c}{a} + \frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 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 ) + 2\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} 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)),x, algorithm="giac")

[Out]

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