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

Optimal. Leaf size=88 \[ -\frac{12 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{35 b^2}-\frac{4 \cos (a+b x)}{35 b^2 \sin ^{\frac{5}{2}}(a+b x)}-\frac{12 \cos (a+b x)}{35 b^2 \sqrt{\sin (a+b x)}}-\frac{2 x}{7 b \sin ^{\frac{7}{2}}(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0443534, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3443, 2636, 2639} \[ -\frac{12 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{35 b^2}-\frac{4 \cos (a+b x)}{35 b^2 \sin ^{\frac{5}{2}}(a+b x)}-\frac{12 \cos (a+b x)}{35 b^2 \sqrt{\sin (a+b x)}}-\frac{2 x}{7 b \sin ^{\frac{7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2636

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

Mathematica [A]  time = 0.299497, size = 73, normalized size = 0.83 \[ -\frac{2 \left (\sin (2 (a+b x))+6 \sin ^3(a+b x) \cos (a+b x)-6 \sin ^{\frac{7}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )+5 b x\right )}{35 b^2 \sin ^{\frac{7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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*Sin[a + b*x]^(7/2))

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{x\cos \left ( bx+a \right ) \left ( \sin \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)/sin(b*x+a)^(9/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos \left (b x + a\right )}{\sin \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)/sin(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)/sin(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)/sin(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)/sin(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 \frac{x \cos \left (b x + a\right )}{\sin \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)/sin(b*x+a)^(9/2),x, algorithm="giac")

[Out]

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