3.3.52 \(\int \sqrt [5]{d \cos (a+b x)} \sin (a+b x) \, dx\) [252]

Optimal. Leaf size=22 \[ -\frac {5 (d \cos (a+b x))^{6/5}}{6 b d} \]

[Out]

-5/6*(d*cos(b*x+a))^(6/5)/b/d

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2645, 30} \begin {gather*} -\frac {5 (d \cos (a+b x))^{6/5}}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(1/5)*Sin[a + b*x],x]

[Out]

(-5*(d*Cos[a + b*x])^(6/5))/(6*b*d)

Rule 30

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

Rule 2645

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 \sqrt [5]{d \cos (a+b x)} \sin (a+b x) \, dx &=-\frac {\text {Subst}\left (\int \sqrt [5]{x} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {5 (d \cos (a+b x))^{6/5}}{6 b d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {5 (d \cos (a+b x))^{6/5}}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(1/5)*Sin[a + b*x],x]

[Out]

(-5*(d*Cos[a + b*x])^(6/5))/(6*b*d)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 19, normalized size = 0.86

method	result	size
derivativedivides	\(-\frac {5 \left (d \cos \left (b x +a \right )\right )^{\frac {6}{5}}}{6 b d}\)	\(19\)
default	\(-\frac {5 \left (d \cos \left (b x +a \right )\right )^{\frac {6}{5}}}{6 b d}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(1/5)*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-5/6*(d*cos(b*x+a))^(6/5)/b/d

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 18, normalized size = 0.82 \begin {gather*} -\frac {5 \, \left (d \cos \left (b x + a\right )\right )^{\frac {6}{5}}}{6 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/5)*sin(b*x+a),x, algorithm="maxima")

[Out]

-5/6*(d*cos(b*x + a))^(6/5)/(b*d)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 21, normalized size = 0.95 \begin {gather*} -\frac {5 \, \left (d \cos \left (b x + a\right )\right )^{\frac {1}{5}} \cos \left (b x + a\right )}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/5)*sin(b*x+a),x, algorithm="fricas")

[Out]

-5/6*(d*cos(b*x + a))^(1/5)*cos(b*x + a)/b

________________________________________________________________________________________

Sympy [A]
time = 2.20, size = 37, normalized size = 1.68 \begin {gather*} \begin {cases} - \frac {5 \sqrt [5]{d \cos {\left (a + b x \right )}} \cos {\left (a + b x \right )}}{6 b} & \text {for}\: b \neq 0 \\x \sqrt [5]{d \cos {\left (a \right )}} \sin {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))**(1/5)*sin(b*x+a),x)

[Out]

Piecewise((-5*(d*cos(a + b*x))**(1/5)*cos(a + b*x)/(6*b), Ne(b, 0)), (x*(d*cos(a))**(1/5)*sin(a), True))

________________________________________________________________________________________

Giac [A]
time = 3.97, size = 21, normalized size = 0.95 \begin {gather*} -\frac {5 \, \left (d \cos \left (b x + a\right )\right )^{\frac {1}{5}} \cos \left (b x + a\right )}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/5)*sin(b*x+a),x, algorithm="giac")

[Out]

-5/6*(d*cos(b*x + a))^(1/5)*cos(b*x + a)/b

________________________________________________________________________________________

Mupad [B]
time = 0.10, size = 18, normalized size = 0.82 \begin {gather*} -\frac {5\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{6/5}}{6\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)*(d*cos(a + b*x))^(1/5),x)

[Out]

-(5*(d*cos(a + b*x))^(6/5))/(6*b*d)

________________________________________________________________________________________