\(\int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx\) [349]

Optimal result

Integrand size = 18, antiderivative size = 65 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b^2}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}} \]

[Out]

4/3*(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)*EllipticE(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))
/b^2-2/3*x/b/sin(b*x+a)^(3/2)-4/3*cos(b*x+a)/b^2/sin(b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3524, 2716, 2719} \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]

[In]

Int[(x*Cos[a + b*x])/Sin[a + b*x]^(5/2),x]

[Out]

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

Rule 2716

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

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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}{3 b \sin ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{3 b} \\ & = -\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 \int \sqrt {\sin (a+b x)} \, dx}{3 b} \\ & = -\frac {4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b^2}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {2 \left (b x-2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {3}{2}}(a+b x)+\sin (2 (a+b x))\right )}{3 b^2 \sin ^{\frac {3}{2}}(a+b x)} \]

[In]

Integrate[(x*Cos[a + b*x])/Sin[a + b*x]^(5/2),x]

[Out]

(-2*(b*x - 2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2) + Sin[2*(a + b*x)]))/(3*b^2*Sin[a + b*x]^(
3/2))

Maple [F]

\[\int \frac {x \cos \left (x b +a \right )}{\sin \left (x b +a \right )^{\frac {5}{2}}}d x\]

[In]

int(x*cos(b*x+a)/sin(b*x+a)^(5/2),x)

[Out]

int(x*cos(b*x+a)/sin(b*x+a)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)**(5/2),x)

[Out]

Integral(x*cos(a + b*x)/sin(a + b*x)**(5/2), x)

Maxima [F]

\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)/sin(b*x + a)^(5/2), x)

Giac [F]

\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x*cos(b*x+a)/sin(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)/sin(b*x + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{5/2}} \,d x \]

[In]

int((x*cos(a + b*x))/sin(a + b*x)^(5/2),x)

[Out]

int((x*cos(a + b*x))/sin(a + b*x)^(5/2), x)