3.348 \(\int \frac{x \cos (a+b x)}{\sin ^{\frac{3}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=38 \[ \frac{4 \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{b^2}-\frac{2 x}{b \sqrt{\sin (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0229804, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3443, 2641} \[ \frac{4 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2}-\frac{2 x}{b \sqrt{\sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cos[a + b*x])/Sin[a + b*x]^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.181046, size = 37, normalized size = 0.97 \[ \frac{2 \left (-2 \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )-\frac{b x}{\sqrt{\sin (a+b x)}}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Cos[a + b*x])/Sin[a + b*x]^(3/2),x]

[Out]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{x\cos \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos{\left (a + b x \right )}}{\sin ^{\frac{3}{2}}{\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)**(3/2),x)

[Out]

Integral(x*cos(a + b*x)/sin(a + b*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)/sin(b*x + a)^(3/2), x)