3.359 \(\int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=108 \[ \frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{147 b^2}+\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]

[Out]

2/7*x/b/csc(b*x+a)^(7/2)+4/49*cos(b*x+a)/b^2/csc(b*x+a)^(5/2)+20/147*cos(b*x+a)/b^2/csc(b*x+a)^(1/2)+20/147*(s
in(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))*csc(b
*x+a)^(1/2)*sin(b*x+a)^(1/2)/b^2

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4213, 3769, 3771, 2641} \[ \frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{147 b^2}+\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/(7*b*Csc[a + b*x]^(7/2)) + (4*Cos[a + b*x])/(49*b^2*Csc[a + b*x]^(5/2)) + (20*Cos[a + b*x])/(147*b^2*Sqr
t[Csc[a + b*x]]) - (20*Sqrt[Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(147*b^2)

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 3769

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4213

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

Rubi steps

\begin {align*} \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx}{7 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx}{49 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {10 \int \sqrt {\csc (a+b x)} \, dx}{147 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {\left (10 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{147 b}\\ &=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{147 b^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.39, size = 93, normalized size = 0.86 \[ \frac {\sqrt {\csc (a+b x)} \left (52 \sin (2 (a+b x))-6 \sin (4 (a+b x))-84 b x \cos (2 (a+b x))+21 b x \cos (4 (a+b x))+80 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+63 b x\right )}{588 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Csc[a + b*x]]*(63*b*x - 84*b*x*Cos[2*(a + b*x)] + 21*b*x*Cos[4*(a + b*x)] + 80*EllipticF[(-2*a + Pi - 2*
b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 52*Sin[2*(a + b*x)] - 6*Sin[4*(a + b*x)]))/(588*b^2)

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x +a \right )}{\csc \left (b x +a \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\cos \left (a+b\,x\right )}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*cos(a + b*x))/(1/sin(a + b*x))^(5/2),x)

[Out]

int((x*cos(a + b*x))/(1/sin(a + b*x))^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos {\left (a + b x \right )}}{\csc ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________