3.345 \(\int \cos ^3(c+d x) (a+a \sin (c+d x))^m \, dx\)

Optimal. Leaf size=55 \[ \frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.0576335, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]

[Out]

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

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

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

Mathematica [A]  time = 0.111585, size = 52, normalized size = 0.95 \[ -\frac{(\sin (c+d x)+1)^2 ((m+2) \sin (c+d x)-m-4) (a (\sin (c+d x)+1))^m}{d (m+2) (m+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]

[Out]

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

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Maple [F]  time = 0.837, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{3} \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)^3*(a+a*sin(d*x+c))^m,x)

[Out]

int(cos(d*x+c)^3*(a+a*sin(d*x+c))^m,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.67177, size = 153, normalized size = 2.78 \begin{align*} \frac{{\left (m \cos \left (d x + c\right )^{2} +{\left ({\left (m + 2\right )} \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) + 4\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{2} + 5 \, d m + 6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

(m*cos(d*x + c)^2 + ((m + 2)*cos(d*x + c)^2 + 4)*sin(d*x + c) + 4)*(a*sin(d*x + c) + a)^m/(d*m^2 + 5*d*m + 6*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)**3*(a+a*sin(d*x+c))**m,x)

[Out]

Timed out

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Giac [B]  time = 1.09057, size = 185, normalized size = 3.36 \begin{align*} -\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 6 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a}{{\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

-((a*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m*m - 2*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a*m + 2*(a
*sin(d*x + c) + a)^3*(a*sin(d*x + c) + a)^m - 6*(a*sin(d*x + c) + a)^2*(a*sin(d*x + c) + a)^m*a)/((a^2*m^2 + 5
*a^2*m + 6*a^2)*a*d)