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

Optimal. Leaf size=24 \[ \frac{(c \sin (a+b x))^{m+1}}{b c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0248876, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2564, 30} \[ \frac{(c \sin (a+b x))^{m+1}}{b c (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0106087, size = 25, normalized size = 1.04 \[ \frac{\sin (a+b x) (c \sin (a+b x))^m}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( c\sin \left ( bx+a \right ) \right ) ^{1+m}}{bc \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.45774, size = 58, normalized size = 2.42 \begin{align*} \frac{\left (c \sin \left (b x + a\right )\right )^{m} \sin \left (b x + a\right )}{b m + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.67533, size = 58, normalized size = 2.42 \begin{align*} \begin{cases} \frac{x \cos{\left (a \right )}}{c \sin{\left (a \right )}} & \text{for}\: b = 0 \wedge m = -1 \\x \left (c \sin{\left (a \right )}\right )^{m} \cos{\left (a \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\sin{\left (a + b x \right )} \right )}}{b c} & \text{for}\: m = -1 \\\frac{c^{m} \sin{\left (a + b x \right )} \sin ^{m}{\left (a + b x \right )}}{b m + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13567, size = 32, normalized size = 1.33 \begin{align*} \frac{\left (c \sin \left (b x + a\right )\right )^{m + 1}}{b c{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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