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

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

[Out]

-4*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))/
b^2-2*x/b/sin(b*x+a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.18, size = 37, normalized size = 0.97 \[ \frac {2 \left (-\frac {b x}{\sqrt {\sin (a+b x)}}-2 F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{b^2} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x +a \right )}{\sin \left (b x +a \right )^{\frac {3}{2}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]

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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