3.204 \(\int \sqrt{d \cos (a+b x)} \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=45 \[ \frac{2 (d \cos (a+b x))^{7/2}}{7 b d^3}-\frac{2 (d \cos (a+b x))^{3/2}}{3 b d} \]

[Out]

(-2*(d*Cos[a + b*x])^(3/2))/(3*b*d) + (2*(d*Cos[a + b*x])^(7/2))/(7*b*d^3)

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Rubi [A]  time = 0.0421826, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2565, 14} \[ \frac{2 (d \cos (a+b x))^{7/2}}{7 b d^3}-\frac{2 (d \cos (a+b x))^{3/2}}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^3,x]

[Out]

(-2*(d*Cos[a + b*x])^(3/2))/(3*b*d) + (2*(d*Cos[a + b*x])^(7/2))/(7*b*d^3)

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sqrt{d \cos (a+b x)} \sin ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \sqrt{x} \left (1-\frac{x^2}{d^2}\right ) \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\sqrt{x}-\frac{x^{5/2}}{d^2}\right ) \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{2 (d \cos (a+b x))^{3/2}}{3 b d}+\frac{2 (d \cos (a+b x))^{7/2}}{7 b d^3}\\ \end{align*}

Mathematica [A]  time = 0.32286, size = 57, normalized size = 1.27 \[ -\frac{d \left (3 \sin ^2(2 (a+b x))+16 \cos ^2(a+b x)-16 \sqrt [4]{\cos ^2(a+b x)}\right )}{42 b \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(16*Cos[a + b*x]^2 - 16*(Cos[a + b*x]^2)^(1/4) + 3*Sin[2*(a + b*x)]^2))/(42*b*Sqrt[d*Cos[a + b*x]])

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Maple [A]  time = 0.043, size = 63, normalized size = 1.4 \begin{align*} -{\frac{8}{21\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d+d} \left ( 6\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}-9\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(1/2)*sin(b*x+a)^3,x)

[Out]

-8/21*(-2*sin(1/2*b*x+1/2*a)^2*d+d)^(1/2)*(6*sin(1/2*b*x+1/2*a)^6-9*sin(1/2*b*x+1/2*a)^4+sin(1/2*b*x+1/2*a)^2+
1)/b

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Maxima [A]  time = 0.973501, size = 49, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (3 \, \left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}} - 7 \, \left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} d^{2}\right )}}{21 \, b d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

2/21*(3*(d*cos(b*x + a))^(7/2) - 7*(d*cos(b*x + a))^(3/2)*d^2)/(b*d^3)

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Fricas [A]  time = 1.8949, size = 88, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (3 \, \cos \left (b x + a\right )^{3} - 7 \, \cos \left (b x + a\right )\right )} \sqrt{d \cos \left (b x + a\right )}}{21 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

2/21*(3*cos(b*x + a)^3 - 7*cos(b*x + a))*sqrt(d*cos(b*x + a))/b

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Sympy [A]  time = 11.4533, size = 65, normalized size = 1.44 \begin{align*} \begin{cases} - \frac{2 \sqrt{d} \sin ^{2}{\left (a + b x \right )} \cos ^{\frac{3}{2}}{\left (a + b x \right )}}{3 b} - \frac{8 \sqrt{d} \cos ^{\frac{7}{2}}{\left (a + b x \right )}}{21 b} & \text{for}\: b \neq 0 \\x \sqrt{d \cos{\left (a \right )}} \sin ^{3}{\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))**(1/2)*sin(b*x+a)**3,x)

[Out]

Piecewise((-2*sqrt(d)*sin(a + b*x)**2*cos(a + b*x)**(3/2)/(3*b) - 8*sqrt(d)*cos(a + b*x)**(7/2)/(21*b), Ne(b,
0)), (x*sqrt(d*cos(a))*sin(a)**3, True))

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Giac [A]  time = 87.9465, size = 49, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (7 \, \left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} - \frac{3 \, \left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}}}{d^{2}}\right )}}{21 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*sin(b*x+a)^3,x, algorithm="giac")

[Out]

-2/21*(7*(d*cos(b*x + a))^(3/2) - 3*(d*cos(b*x + a))^(7/2)/d^2)/(b*d)