3.357 \(\int (d \cos (a+b x))^n \sin (a+b x) \, dx\)

Optimal. Leaf size=25 \[ -\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)} \]

[Out]

-((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n)))

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Rubi [A]  time = 0.022271, antiderivative size = 25, 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 = {2565, 30} \[ -\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n)))

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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 (d \cos (a+b x))^n \sin (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^n \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{(d \cos (a+b x))^{1+n}}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0100037, size = 26, normalized size = 1.04 \[ -\frac{\cos (a+b x) (d \cos (a+b x))^n}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Cos[a + b*x]*(d*Cos[a + b*x])^n)/(b*(1 + n)))

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Maple [A]  time = 0.003, size = 26, normalized size = 1. \begin{align*} -{\frac{ \left ( d\cos \left ( bx+a \right ) \right ) ^{1+n}}{bd \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*cos(b*x+a))^(1+n)/b/d/(1+n)

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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((d*cos(b*x+a))^n*sin(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.2406, size = 59, normalized size = 2.36 \begin{align*} -\frac{\left (d \cos \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )}{b n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*cos(b*x + a))^n*cos(b*x + a)/(b*n + b)

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Sympy [A]  time = 1.80926, size = 61, normalized size = 2.44 \begin{align*} \begin{cases} \frac{x \sin{\left (a \right )}}{d \cos{\left (a \right )}} & \text{for}\: b = 0 \wedge n = -1 \\x \left (d \cos{\left (a \right )}\right )^{n} \sin{\left (a \right )} & \text{for}\: b = 0 \\- \frac{\log{\left (\cos{\left (a + b x \right )} \right )}}{b d} & \text{for}\: n = -1 \\- \frac{d^{n} \cos{\left (a + b x \right )} \cos ^{n}{\left (a + b x \right )}}{b n + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15563, size = 34, normalized size = 1.36 \begin{align*} -\frac{\left (d \cos \left (b x + a\right )\right )^{n + 1}}{b d{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*cos(b*x + a))^(n + 1)/(b*d*(n + 1))