3.219 \(\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=67 \[ \frac{(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac{(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac{(a \sin (c+d x)+a)^5}{5 a d} \]

[Out]

(a + a*Sin[c + d*x])^5/(5*a*d) - (a + a*Sin[c + d*x])^6/(3*a^2*d) + (a + a*Sin[c + d*x])^7/(7*a^3*d)

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Rubi [A]  time = 0.0753, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac{(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac{(a \sin (c+d x)+a)^5}{5 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]

[Out]

(a + a*Sin[c + d*x])^5/(5*a*d) - (a + a*Sin[c + d*x])^6/(3*a^2*d) + (a + a*Sin[c + d*x])^7/(7*a^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 ^2(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 (a+x)^4}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 (a+x)^4-2 a (a+x)^5+(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{(a+a \sin (c+d x))^5}{5 a d}-\frac{(a+a \sin (c+d x))^6}{3 a^2 d}+\frac{(a+a \sin (c+d x))^7}{7 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.332869, size = 80, normalized size = 1.19 \[ -\frac{a^4 (-7245 \sin (c+d x)+3395 \sin (3 (c+d x))-609 \sin (5 (c+d x))+15 \sin (7 (c+d x))+5460 \cos (2 (c+d x))-1680 \cos (4 (c+d x))+140 \cos (6 (c+d x))-630)}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]

[Out]

-(a^4*(-630 + 5460*Cos[2*(c + d*x)] - 1680*Cos[4*(c + d*x)] + 140*Cos[6*(c + d*x)] - 7245*Sin[c + d*x] + 3395*
Sin[3*(c + d*x)] - 609*Sin[5*(c + d*x)] + 15*Sin[7*(c + d*x)]))/(6720*d)

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Maple [A]  time = 0.022, size = 70, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{2\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3}}+{\frac{6\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x)

[Out]

1/d*(1/7*a^4*sin(d*x+c)^7+2/3*a^4*sin(d*x+c)^6+6/5*a^4*sin(d*x+c)^5+a^4*sin(d*x+c)^4+1/3*a^4*sin(d*x+c)^3)

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Maxima [A]  time = 1.12522, size = 96, normalized size = 1.43 \begin{align*} \frac{15 \, a^{4} \sin \left (d x + c\right )^{7} + 70 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5} + 105 \, a^{4} \sin \left (d x + c\right )^{4} + 35 \, a^{4} \sin \left (d x + c\right )^{3}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/105*(15*a^4*sin(d*x + c)^7 + 70*a^4*sin(d*x + c)^6 + 126*a^4*sin(d*x + c)^5 + 105*a^4*sin(d*x + c)^4 + 35*a^
4*sin(d*x + c)^3)/d

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Fricas [A]  time = 1.95417, size = 247, normalized size = 3.69 \begin{align*} -\frac{70 \, a^{4} \cos \left (d x + c\right )^{6} - 315 \, a^{4} \cos \left (d x + c\right )^{4} + 420 \, a^{4} \cos \left (d x + c\right )^{2} +{\left (15 \, a^{4} \cos \left (d x + c\right )^{6} - 171 \, a^{4} \cos \left (d x + c\right )^{4} + 332 \, a^{4} \cos \left (d x + c\right )^{2} - 176 \, a^{4}\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/105*(70*a^4*cos(d*x + c)^6 - 315*a^4*cos(d*x + c)^4 + 420*a^4*cos(d*x + c)^2 + (15*a^4*cos(d*x + c)^6 - 171
*a^4*cos(d*x + c)^4 + 332*a^4*cos(d*x + c)^2 - 176*a^4)*sin(d*x + c))/d

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Sympy [A]  time = 10.2798, size = 119, normalized size = 1.78 \begin{align*} \begin{cases} \frac{a^{4} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac{2 a^{4} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac{6 a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{2 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{a^{4} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{4} \sin ^{2}{\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)**2*(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((a**4*sin(c + d*x)**7/(7*d) + 2*a**4*sin(c + d*x)**6/(3*d) + 6*a**4*sin(c + d*x)**5/(5*d) + a**4*sin
(c + d*x)**3/(3*d) - 2*a**4*sin(c + d*x)**2*cos(c + d*x)**2/d - a**4*cos(c + d*x)**4/d, Ne(d, 0)), (x*(a*sin(c
) + a)**4*sin(c)**2*cos(c), True))

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Giac [A]  time = 1.18746, size = 96, normalized size = 1.43 \begin{align*} \frac{15 \, a^{4} \sin \left (d x + c\right )^{7} + 70 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5} + 105 \, a^{4} \sin \left (d x + c\right )^{4} + 35 \, a^{4} \sin \left (d x + c\right )^{3}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/105*(15*a^4*sin(d*x + c)^7 + 70*a^4*sin(d*x + c)^6 + 126*a^4*sin(d*x + c)^5 + 105*a^4*sin(d*x + c)^4 + 35*a^
4*sin(d*x + c)^3)/d