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

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

[Out]

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

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Rubi [A]  time = 0.026876, 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 = {2668, 32} \[ \frac{(a+b \sin (c+d x))^{m+1}}{b d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A]  time = 0.0248363, size = 26, normalized size = 1. \[ \frac{(a+b \sin (c+d x))^{m+1}}{b d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{1+m}}{bd \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a+b*sin(d*x+c))^(1+m)/b/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+b*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40809, size = 80, normalized size = 3.08 \begin{align*} \frac{{\left (b \sin \left (d x + c\right ) + a\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b d m + b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.64656, size = 99, normalized size = 3.81 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \wedge m = -1 \\\frac{a^{m} \sin{\left (c + d x \right )}}{d} & \text{for}\: b = 0 \\x \left (a + b \sin{\left (c \right )}\right )^{m} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{\log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{b d} & \text{for}\: m = -1 \\\frac{a \left (a + b \sin{\left (c + d x \right )}\right )^{m}}{b d m + b d} + \frac{b \left (a + b \sin{\left (c + d x \right )}\right )^{m} \sin{\left (c + d x \right )}}{b d m + b d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10451, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b d{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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