3.352 \(\int x \cos (a+b x) \csc ^{\frac{9}{2}}(a+b x) \, dx\)

Optimal. Leaf size=108 \[ -\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{12 \cos (a+b x) \sqrt{\csc (a+b x)}}{35 b^2}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b} \]

[Out]

(-12*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(35*b^2) - (4*Cos[a + b*x]*Csc[a + b*x]^(5/2))/(35*b^2) - (2*x*Csc[a + b
*x]^(7/2))/(7*b) - (12*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(35*b^2)

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Rubi [A]  time = 0.0576411, antiderivative size = 108, normalized size of antiderivative = 1., 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, 3768, 3771, 2639} \[ -\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{12 \cos (a+b x) \sqrt{\csc (a+b x)}}{35 b^2}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*x]*Csc[a + b*x]^(9/2),x]

[Out]

(-12*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(35*b^2) - (4*Cos[a + b*x]*Csc[a + b*x]^(5/2))/(35*b^2) - (2*x*Csc[a + b
*x]^(7/2))/(7*b) - (12*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(35*b^2)

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]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 2639

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

Rubi steps

\begin{align*} \int x \cos (a+b x) \csc ^{\frac{9}{2}}(a+b x) \, dx &=-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b}+\frac{2 \int \csc ^{\frac{7}{2}}(a+b x) \, dx}{7 b}\\ &=-\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b}+\frac{6 \int \csc ^{\frac{3}{2}}(a+b x) \, dx}{35 b}\\ &=-\frac{12 \cos (a+b x) \sqrt{\csc (a+b x)}}{35 b^2}-\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{6 \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx}{35 b}\\ &=-\frac{12 \cos (a+b x) \sqrt{\csc (a+b x)}}{35 b^2}-\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{\left (6 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx}{35 b}\\ &=-\frac{12 \cos (a+b x) \sqrt{\csc (a+b x)}}{35 b^2}-\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{12 \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{35 b^2}\\ \end{align*}

Mathematica [A]  time = 0.272145, size = 73, normalized size = 0.68 \[ -\frac{2 \csc ^{\frac{7}{2}}(a+b x) \left (\sin (2 (a+b x))+6 \sin ^3(a+b x) \cos (a+b x)-6 \sin ^{\frac{7}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )+5 b x\right )}{35 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]*Csc[a + b*x]^(9/2),x]

[Out]

(-2*Csc[a + b*x]^(7/2)*(5*b*x + 6*Cos[a + b*x]*Sin[a + b*x]^3 - 6*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a +
b*x]^(7/2) + Sin[2*(a + b*x)]))/(35*b^2)

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Maple [F]  time = 0.116, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{{\frac{9}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(b*x+a)*csc(b*x+a)^(9/2),x)

[Out]

int(x*cos(b*x+a)*csc(b*x+a)^(9/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)*csc(b*x + a)^(9/2), 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)*csc(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \csc \left (b x + a\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)*csc(b*x+a)^(9/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)*csc(b*x + a)^(9/2), x)