3.191 \(\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=33 \[ \frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0358782, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2833, 12, 43} \[ \frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.090569, size = 30, normalized size = 0.91 \[ \frac{4 a \sin ^3(c+d x)-3 a \cos (2 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cos[2*(c + d*x)] + 4*a*Sin[c + d*x]^3)/(12*d)

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Maple [A]  time = 0.013, size = 28, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}a}{3}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2}} \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)),x)

[Out]

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

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Maxima [A]  time = 1.29919, size = 38, normalized size = 1.15 \begin{align*} \frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2}}{6 \, 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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64546, size = 93, normalized size = 2.82 \begin{align*} -\frac{3 \, a \cos \left (d x + c\right )^{2} + 2 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right )}{6 \, 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)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.3498, size = 41, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{a \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin{\left (c \right )} \cos{\left (c \right )} & \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)),x)

[Out]

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

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Giac [A]  time = 1.29005, size = 38, normalized size = 1.15 \begin{align*} \frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2}}{6 \, 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)),x, algorithm="giac")

[Out]

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