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

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

[Out]

(-5*(d*Cos[a + b*x])^(6/5))/(6*b*d)

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Rubi [A]  time = 0.0222108, antiderivative size = 22, normalized size of antiderivative = 1., 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 = {2565, 30} \[ -\frac{5 (d \cos (a+b x))^{6/5}}{6 b d} \]

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 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])

Rule 30

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

Rubi steps

\begin{align*} \int \sqrt [5]{d \cos (a+b x)} \sin (a+b x) \, dx &=-\frac{\operatorname{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 = 0.0201343, size = 22, normalized size = 1. \[ -\frac{5 (d \cos (a+b x))^{6/5}}{6 b d} \]

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)

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Maple [A]  time = 0.004, size = 19, normalized size = 0.9 \begin{align*} -{\frac{5}{6\,bd} \left ( d\cos \left ( bx+a \right ) \right ) ^{{\frac{6}{5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.977107, size = 24, normalized size = 1.09 \begin{align*} -\frac{5 \, \left (d \cos \left (b x + a\right )\right )^{\frac{6}{5}}}{6 \, b d} \end{align*}

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)

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Fricas [A]  time = 2.20784, size = 59, normalized size = 2.68 \begin{align*} -\frac{5 \, \left (d \cos \left (b x + a\right )\right )^{\frac{1}{5}} \cos \left (b x + a\right )}{6 \, b} \end{align*}

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

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Sympy [A]  time = 33.7158, size = 34, normalized size = 1.55 \begin{align*} \begin{cases} - \frac{5 \sqrt [5]{d} \cos ^{\frac{6}{5}}{\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{align*}

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

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Giac [A]  time = 1.23505, size = 28, normalized size = 1.27 \begin{align*} -\frac{5 \, \left (d \cos \left (b x + a\right )\right )^{\frac{1}{5}} \cos \left (b x + a\right )}{6 \, b} \end{align*}

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