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

Optimal. Leaf size=19 \[ \frac{(a+b \sin (x))^{n+1}}{b (n+1)} \]

[Out]

(a + b*Sin[x])^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0216845, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2668, 32} \[ \frac{(a+b \sin (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*(a + b*Sin[x])^n,x]

[Out]

(a + b*Sin[x])^(1 + n)/(b*(1 + n))

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

Mathematica [A]  time = 0.0204927, size = 18, normalized size = 0.95 \[ \frac{(a+b \sin (x))^{n+1}}{b n+b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*(a + b*Sin[x])^n,x]

[Out]

(a + b*Sin[x])^(1 + n)/(b + b*n)

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Maple [A]  time = 0.007, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( a+b\sin \left ( x \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.06725, size = 58, normalized size = 3.05 \begin{align*} \frac{{\left (b \sin \left (x\right ) + a\right )}{\left (b \sin \left (x\right ) + a\right )}^{n}}{b n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*sin(x) + a)*(b*sin(x) + a)^n/(b*n + b)

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Sympy [A]  time = 2.56974, size = 56, normalized size = 2.95 \begin{align*} \begin{cases} \frac{\sin{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = -1 \\a^{n} \sin{\left (x \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + \sin{\left (x \right )} \right )}}{b} & \text{for}\: n = -1 \\\frac{a \left (a + b \sin{\left (x \right )}\right )^{n}}{b n + b} + \frac{b \left (a + b \sin{\left (x \right )}\right )^{n} \sin{\left (x \right )}}{b n + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08492, size = 26, normalized size = 1.37 \begin{align*} \frac{{\left (b \sin \left (x\right ) + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*sin(x) + a)^(n + 1)/(b*(n + 1))