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

Optimal. Leaf size=26 \[ \frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]

[Out]

(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))

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Rubi [A]  time = 0.0275411, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + 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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0321921, size = 26, normalized size = 1. \[ \frac{(a (\sin (c+d x)+1))^{m+1}}{a d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(1 + Sin[c + d*x]))^(1 + m)/(a*d*(1 + m))

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Maple [A]  time = 0.001, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{1+m}}{da \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.29077, size = 72, normalized size = 2.77 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}{\left (\sin \left (d x + c\right ) + 1\right )}}{d m + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.64415, size = 80, normalized size = 3.08 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{for}\: d = 0 \wedge m = -1 \\x \left (a \sin{\left (c \right )} + a\right )^{m} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} & \text{for}\: m = -1 \\\frac{\left (a \sin{\left (c + d x \right )} + a\right )^{m} \sin{\left (c + d x \right )}}{d m + d} + \frac{\left (a \sin{\left (c + d x \right )} + a\right )^{m}}{d m + d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*cos(c)/(a*sin(c) + a), Eq(d, 0) & Eq(m, -1)), (x*(a*sin(c) + a)**m*cos(c), Eq(d, 0)), (log(sin(c
+ d*x) + 1)/(a*d), Eq(m, -1)), ((a*sin(c + d*x) + a)**m*sin(c + d*x)/(d*m + d) + (a*sin(c + d*x) + a)**m/(d*m
+ d), True))

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Giac [A]  time = 1.07156, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m + 1}}{a d{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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