3.358 \(\int \frac{x \cos (a+b x)}{\csc ^{\frac{3}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=85 \[ \frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)} \]

[Out]

(2*x)/(5*b*Csc[a + b*x]^(5/2)) + (4*Cos[a + b*x])/(25*b^2*Csc[a + b*x]^(3/2)) - (12*Sqrt[Csc[a + b*x]]*Ellipti
cE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(25*b^2)

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Rubi [A]  time = 0.0437454, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4213, 3769, 3771, 2639} \[ \frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cos[a + b*x])/Csc[a + b*x]^(3/2),x]

[Out]

(2*x)/(5*b*Csc[a + b*x]^(5/2)) + (4*Cos[a + b*x])/(25*b^2*Csc[a + b*x]^(3/2)) - (12*Sqrt[Csc[a + b*x]]*Ellipti
cE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(25*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 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 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)}{\csc ^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{2 \int \frac{1}{\csc ^{\frac{5}{2}}(a+b x)} \, dx}{5 b}\\ &=\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\frac{6 \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx}{25 b}\\ &=\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\frac{\left (6 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx}{25 b}\\ &=\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\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)}}{25 b^2}\\ \end{align*}

Mathematica [C]  time = 0.997851, size = 114, normalized size = 1.34 \[ \frac{\tan \left (\frac{1}{2} (a+b x)\right ) \left (4 \sqrt{2} \sqrt{\frac{1}{\cos (a+b x)+1}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )+10 b x \sin (a+b x)+5 b x \sin (2 (a+b x))+4 \cos (a+b x)+2 \cos (2 (a+b x))-10\right )}{25 b^2 \sqrt{\csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Cos[a + b*x])/Csc[a + b*x]^(3/2),x]

[Out]

((-10 + 4*Cos[a + b*x] + 2*Cos[2*(a + b*x)] + 4*Sqrt[2]*Sqrt[(1 + Cos[a + b*x])^(-1)]*Hypergeometric2F1[1/2, 3
/4, 7/4, -Tan[(a + b*x)/2]^2] + 10*b*x*Sin[a + b*x] + 5*b*x*Sin[2*(a + b*x)])*Tan[(a + b*x)/2])/(25*b^2*Sqrt[C
sc[a + b*x]])

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{x\cos \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

int(x*cos(b*x+a)/csc(b*x+a)^(3/2),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 )}{\csc \left (b x + a\right )^{\frac{3}{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)^(3/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)/csc(b*x + a)^(3/2), 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)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos{\left (a + b x \right )}}{\csc ^{\frac{3}{2}}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)/csc(b*x+a)**(3/2),x)

[Out]

Integral(x*cos(a + b*x)/csc(a + b*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac{3}{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)^(3/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)/csc(b*x + a)^(3/2), x)