3.42 \(\int \cos ^3(a+b x) \sin (a+b x) \, dx\)

Optimal. Leaf size=15 \[ -\frac {\cos ^4(a+b x)}{4 b} \]

[Out]

-1/4*cos(b*x+a)^4/b

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2565, 30} \[ -\frac {\cos ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3*Sin[a + b*x],x]

[Out]

-Cos[a + b*x]^4/(4*b)

Rule 30

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

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])

Rubi steps

\begin {align*} \int \cos ^3(a+b x) \sin (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^4(a+b x)}{4 b}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {\cos ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^3*Sin[a + b*x],x]

[Out]

-1/4*Cos[a + b*x]^4/b

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fricas [A]  time = 0.42, size = 13, normalized size = 0.87 \[ -\frac {\cos \left (b x + a\right )^{4}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/4*cos(b*x + a)^4/b

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giac [A]  time = 0.21, size = 24, normalized size = 1.60 \[ -\frac {\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a),x, algorithm="giac")

[Out]

-1/4*(sin(b*x + a)^4 - 2*sin(b*x + a)^2)/b

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maple [A]  time = 0.02, size = 14, normalized size = 0.93 \[ -\frac {\cos ^{4}\left (b x +a \right )}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3*sin(b*x+a),x)

[Out]

-1/4*cos(b*x+a)^4/b

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maxima [A]  time = 0.33, size = 13, normalized size = 0.87 \[ -\frac {\cos \left (b x + a\right )^{4}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/4*cos(b*x + a)^4/b

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mupad [B]  time = 0.06, size = 13, normalized size = 0.87 \[ -\frac {{\cos \left (a+b\,x\right )}^4}{4\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^3*sin(a + b*x),x)

[Out]

-cos(a + b*x)^4/(4*b)

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sympy [A]  time = 1.37, size = 22, normalized size = 1.47 \[ \begin {cases} - \frac {\cos ^{4}{\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x \sin {\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**3*sin(b*x+a),x)

[Out]

Piecewise((-cos(a + b*x)**4/(4*b), Ne(b, 0)), (x*sin(a)*cos(a)**3, True))

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