3.1.70 \(\int \cos (a+b x) \sin ^3(a+b x) \, dx\) [70]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {2644, 30} \begin {gather*} \frac {\sin ^4(a+b x)}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Sin[a + b*x]^3,x]

[Out]

Sin[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 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) \sin ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int x^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^4(a+b x)}{4 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\sin ^4(a+b x)}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[a + b*x]^4/(4*b)

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Maple [A]
time = 0.04, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\sin ^{4}\left (b x +a \right )}{4 b}\)	\(14\)
default	\(\frac {\sin ^{4}\left (b x +a \right )}{4 b}\)	\(14\)
risch	\(\frac {\cos \left (4 b x +4 a \right )}{32 b}-\frac {\cos \left (2 b x +2 a \right )}{8 b}\)	\(30\)
norman	\(\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} \frac {\sin \left (b x + a\right )^{4}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sin(b*x + a)^4/b

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Fricas [A]
time = 0.37, size = 24, normalized size = 1.60 \begin {gather*} \frac {\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.15, size = 20, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {\sin ^{4}{\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.47, size = 13, normalized size = 0.87 \begin {gather*} \frac {\sin \left (b x + a\right )^{4}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sin(b*x + a)^4/b

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Mupad [B]
time = 0.37, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\sin \left (a+b\,x\right )}^4}{4\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)*sin(a + b*x)^3,x)

[Out]

sin(a + b*x)^4/(4*b)

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