3.4.42 \(\int \cos (a+b x) (c \sin (a+b x))^m \, dx\) [342]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {2644, 30} \begin {gather*} \frac {(c \sin (a+b x))^{m+1}}{b c (m+1)} \end {gather*}

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 30

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

Rule 2644

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])

Rubi steps

\begin {align*} \int \cos (a+b x) (c \sin (a+b x))^m \, dx &=\frac {\text {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*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.04 \begin {gather*} \frac {\sin (a+b x) (c \sin (a+b x))^m}{b (1+m)} \end {gather*}

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 = 1.11, size = 25, normalized size = 1.04

method	result	size
derivativedivides	\(\frac {\left (c \sin \left (b x +a \right )\right )^{1+m}}{b c \left (1+m \right )}\)	\(25\)
default	\(\frac {\left (c \sin \left (b x +a \right )\right )^{1+m}}{b c \left (1+m \right )}\)	\(25\)
norman	\(\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) {\mathrm e}^{m \ln \left (\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}}{b \left (1+m \right ) \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\)	\(66\)
risch	\(\text {Expression too large to display}\)	\(1016\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (c \sin \left (b x + a\right )\right )^{m + 1}}{b c {\left (m + 1\right )}} \end {gather*}

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]

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

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Fricas [A]
time = 0.41, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (c \sin \left (b x + a\right )\right )^{m} \sin \left (b x + a\right )}{b m + b} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).
time = 0.46, size = 56, normalized size = 2.33 \begin {gather*} \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 {\left (c \sin {\left (a + b x \right )}\right )^{m} \sin {\left (a + b x \right )}}{b m + b} & \text {otherwise} \end {cases} \end {gather*}

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*sin(a + b*x))**m*sin(a + b*x)/(b*m + b), True))

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Giac [A]
time = 3.24, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (c \sin \left (b x + a\right )\right )^{m + 1}}{b c {\left (m + 1\right )}} \end {gather*}

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))

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Mupad [B]
time = 0.61, size = 25, normalized size = 1.04 \begin {gather*} \frac {\sin \left (a+b\,x\right )\,{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{b\,\left (m+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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