3.346 \(\int x \cos (a+b x) \sqrt{\sin (a+b x)} \, dx\)

Optimal. Leaf size=65 \[ -\frac{4 \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{9 b^2}+\frac{4 \sqrt{\sin (a+b x)} \cos (a+b x)}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b} \]

[Out]

(-4*EllipticF[(a - Pi/2 + b*x)/2, 2])/(9*b^2) + (4*Cos[a + b*x]*Sqrt[Sin[a + b*x]])/(9*b^2) + (2*x*Sin[a + b*x
]^(3/2))/(3*b)

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Rubi [A]  time = 0.0315743, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3443, 2635, 2641} \[ -\frac{4 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{9 b^2}+\frac{4 \sqrt{\sin (a+b x)} \cos (a+b x)}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*EllipticF[(a - Pi/2 + b*x)/2, 2])/(9*b^2) + (4*Cos[a + b*x]*Sqrt[Sin[a + b*x]])/(9*b^2) + (2*x*Sin[a + b*x
]^(3/2))/(3*b)

Rule 3443

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int x \cos (a+b x) \sqrt{\sin (a+b x)} \, dx &=\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \sin ^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=\frac{4 \cos (a+b x) \sqrt{\sin (a+b x)}}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{9 b}\\ &=-\frac{4 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{9 b^2}+\frac{4 \cos (a+b x) \sqrt{\sin (a+b x)}}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.188767, size = 56, normalized size = 0.86 \[ \frac{4 \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+2 \sqrt{\sin (a+b x)} (3 b x \sin (a+b x)+2 \cos (a+b x))}{9 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + 2*Sqrt[Sin[a + b*x]]*(2*Cos[a + b*x] + 3*b*x*Sin[a + b*x]))/(9*b^2)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \sqrt{\sin \left ( bx+a \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sqrt{\sin \left (b x + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\sin{\left (a + b x \right )}} \cos{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*sin(b*x+a)**(1/2),x)

[Out]

Integral(x*sqrt(sin(a + b*x))*cos(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sqrt{\sin \left (b x + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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