3.928 \(\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx\)

Optimal. Leaf size=134 \[ -\frac{4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac{6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]

[Out]

(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m)) - (4*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) + (6*(a + a*Sin[
c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*Sin[c + d*x])^(
5 + m)/(a^5*d*(5 + m))

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Rubi [A]  time = 0.119252, antiderivative size = 134, 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} \[ -\frac{4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac{6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m)) - (4*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) + (6*(a + a*Sin[
c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*Sin[c + d*x])^(
5 + m)/(a^5*d*(5 + m))

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 ^4(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 (a+x)^m}{a^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x^4 (a+x)^m \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^4 (a+x)^m-4 a^3 (a+x)^{1+m}+6 a^2 (a+x)^{2+m}-4 a (a+x)^{3+m}+(a+x)^{4+m}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{(a+a \sin (c+d x))^{1+m}}{a d (1+m)}-\frac{4 (a+a \sin (c+d x))^{2+m}}{a^2 d (2+m)}+\frac{6 (a+a \sin (c+d x))^{3+m}}{a^3 d (3+m)}-\frac{4 (a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}+\frac{(a+a \sin (c+d x))^{5+m}}{a^5 d (5+m)}\\ \end{align*}

Mathematica [A]  time = 1.40365, size = 150, normalized size = 1.12 \[ \frac{(a (\sin (c+d x)+1))^{m+1} \left (\frac{3 \left (-2 \left (m^2+3 m+2\right ) \cos (2 (c+d x))-8 (m+1) \sin (c+d x)+m^2+m+6\right )}{(m+1) (m+2) (m+3)}+\frac{16 (\sin (c+d x)+1)^4}{m+5}-\frac{64 (\sin (c+d x)+1)^3}{m+4}+\frac{84 (\sin (c+d x)+1)^2}{m+3}-\frac{40 (\sin (c+d x)+1)}{m+2}+\frac{7}{m+1}\right )}{16 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(1 + Sin[c + d*x]))^(1 + m)*(7/(1 + m) - (40*(1 + Sin[c + d*x]))/(2 + m) + (84*(1 + Sin[c + d*x])^2)/(3 +
m) - (64*(1 + Sin[c + d*x])^3)/(4 + m) + (16*(1 + Sin[c + d*x])^4)/(5 + m) + (3*(6 + m + m^2 - 2*(2 + 3*m + m^
2)*Cos[2*(c + d*x)] - 8*(1 + m)*Sin[c + d*x]))/((1 + m)*(2 + m)*(3 + m))))/(16*a*d)

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Maple [F]  time = 2.483, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07683, size = 215, normalized size = 1.6 \begin{align*} \frac{{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (d x + c\right )^{5} +{\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (d x + c\right )^{4} - 4 \,{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} + 12 \,{\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} - 24 \, a^{m} m \sin \left (d x + c\right ) + 24 \, a^{m}\right )}{\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*a^m*sin(d*x + c)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(d*x + c)^4 - 4*
(m^3 + 3*m^2 + 2*m)*a^m*sin(d*x + c)^3 + 12*(m^2 + m)*a^m*sin(d*x + c)^2 - 24*a^m*m*sin(d*x + c) + 24*a^m)*(si
n(d*x + c) + 1)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*d)

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Fricas [A]  time = 1.85389, size = 501, normalized size = 3.74 \begin{align*} \frac{{\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} \cos \left (d x + c\right )^{4} + m^{4} + 6 \, m^{3} - 2 \,{\left (m^{4} + 6 \, m^{3} + 17 \, m^{2} + 12 \, m\right )} \cos \left (d x + c\right )^{2} + 23 \, m^{2} +{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} \cos \left (d x + c\right )^{4} + m^{4} + 6 \, m^{3} - 2 \,{\left (m^{4} + 8 \, m^{3} + 29 \, m^{2} + 46 \, m + 24\right )} \cos \left (d x + c\right )^{2} + 23 \, m^{2} + 18 \, m + 24\right )} \sin \left (d x + c\right ) + 18 \, m + 24\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{5} + 15 \, d m^{4} + 85 \, d m^{3} + 225 \, d m^{2} + 274 \, d m + 120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((m^4 + 6*m^3 + 11*m^2 + 6*m)*cos(d*x + c)^4 + m^4 + 6*m^3 - 2*(m^4 + 6*m^3 + 17*m^2 + 12*m)*cos(d*x + c)^2 +
23*m^2 + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*cos(d*x + c)^4 + m^4 + 6*m^3 - 2*(m^4 + 8*m^3 + 29*m^2 + 46*m +
24)*cos(d*x + c)^2 + 23*m^2 + 18*m + 24)*sin(d*x + c) + 18*m + 24)*(a*sin(d*x + c) + a)^m/(d*m^5 + 15*d*m^4 +
85*d*m^3 + 225*d*m^2 + 274*d*m + 120*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)**4*(a+a*sin(d*x+c))**m,x)

[Out]

Timed out

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Giac [B]  time = 1.27006, size = 1068, normalized size = 7.97 \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)*sin(d*x+c)^4*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

((a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*m^4 - 4*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a*m^4 + 6
*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*a^2*m^4 - 4*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a^3*m
^4 + (a*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^m*a^4*m^4 + 10*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*m^
3 - 44*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a*m^3 + 72*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*
a^2*m^3 - 52*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a^3*m^3 + 14*(a*sin(d*x + c) + a)*(a*sin(d*x + c) +
a)^m*a^4*m^3 + 35*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m*m^2 - 164*(a*sin(d*x + c) + a)^4*(a*sin(d*x +
c) + a)^m*a*m^2 + 294*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*a^2*m^2 - 236*(a*sin(d*x + c) + a)^2*(a*s
in(d*x + c) + a)^m*a^3*m^2 + 71*(a*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^m*a^4*m^2 + 50*(a*sin(d*x + c) + a)^
5*(a*sin(d*x + c) + a)^m*m - 244*(a*sin(d*x + c) + a)^4*(a*sin(d*x + c) + a)^m*a*m + 468*(a*sin(d*x + c) + a)^
3*(a*sin(d*x + c) + a)^m*a^2*m - 428*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a^3*m + 154*(a*sin(d*x + c)
+ a)*(a*sin(d*x + c) + a)^m*a^4*m + 24*(a*sin(d*x + c) + a)^5*(a*sin(d*x + c) + a)^m - 120*(a*sin(d*x + c) +
a)^4*(a*sin(d*x + c) + a)^m*a + 240*(a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*a^2 - 240*(a*sin(d*x + c) +
a)^2*(a*sin(d*x + c) + a)^m*a^3 + 120*(a*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^m*a^4)/((a^4*m^5 + 15*a^4*m^4
+ 85*a^4*m^3 + 225*a^4*m^2 + 274*a^4*m + 120*a^4)*a*d)