\(\int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx\) [348]

Optimal result

Integrand size = 18, antiderivative size = 38 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \]

[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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3524, 2720} \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \]

[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 2720

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

Rule 3524

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*} \text {integral}& = -\frac {2 x}{b \sqrt {\sin (a+b x)}}+\frac {2 \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{b} \\ & = \frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \left (-2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )-\frac {b x}{\sqrt {\sin (a+b x)}}\right )}{b^2} \]

[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

Maple [F]

\[\int \frac {x \cos \left (x b +a \right )}{\sin \left (x b +a \right )^{\frac {3}{2}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{3/2}} \,d x \]

[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)