3.356 \(\int x \cos (a+b x) \sqrt{\csc (a+b x)} \, dx\)

Optimal. Leaf size=58 \[ \frac{2 x}{b \sqrt{\csc (a+b x)}}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0319711, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4213, 3771, 2639} \[ \frac{2 x}{b \sqrt{\csc (a+b x)}}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]

[Out]

(2*x)/(b*Sqrt[Csc[a + b*x]]) - (4*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/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 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) \sqrt{\csc (a+b x)} \, dx &=\frac{2 x}{b \sqrt{\csc (a+b x)}}-\frac{2 \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx}{b}\\ &=\frac{2 x}{b \sqrt{\csc (a+b x)}}-\frac{\left (2 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx}{b}\\ &=\frac{2 x}{b \sqrt{\csc (a+b x)}}-\frac{4 \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)}}{b^2}\\ \end{align*}

Mathematica [C]  time = 0.729526, size = 106, normalized size = 1.83 \[ \frac{4 \sin \left (\frac{1}{2} (a+b x)\right ) \cos \left (\frac{1}{2} (a+b x)\right ) \sqrt{\csc (a+b x)} \left (2 \tan \left (\frac{1}{2} (a+b x)\right ) \sqrt{\sec ^2\left (\frac{1}{2} (a+b x)\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-6 \tan \left (\frac{1}{2} (a+b x)\right )+3 b x\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]

[Out]

(4*Cos[(a + b*x)/2]*Sqrt[Csc[a + b*x]]*Sin[(a + b*x)/2]*(3*b*x - 6*Tan[(a + b*x)/2] + 2*Hypergeometric2F1[1/2,
3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sqrt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(3*b^2)

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Maple [C]  time = 0.212, size = 308, normalized size = 5.3 \begin{align*}{\frac{-i \left ( bx+2\,i \right ) \left ( \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}-1 \right ) \sqrt{2}}{{b}^{2}{{\rm e}^{i \left ( bx+a \right ) }}}\sqrt{{\frac{i{{\rm e}^{i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}-1}}}}-2\,{\frac{\sqrt{2}\sqrt{i \left ( \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}-1 \right ){{\rm e}^{i \left ( bx+a \right ) }}}}{{b}^{2}{{\rm e}^{i \left ( bx+a \right ) }}} \left ({\frac{-2\,i \left ( -i+i \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2} \right ) }{\sqrt{{{\rm e}^{i \left ( bx+a \right ) }} \left ( -i+i \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2} \right ) }}}-{\frac{\sqrt{{{\rm e}^{i \left ( bx+a \right ) }}+1}\sqrt{-2\,{{\rm e}^{i \left ( bx+a \right ) }}+2}\sqrt{-{{\rm e}^{i \left ( bx+a \right ) }}} \left ( -2\,{\it EllipticE} \left ( \sqrt{{{\rm e}^{i \left ( bx+a \right ) }}+1},1/2\,\sqrt{2} \right ) +{\it EllipticF} \left ( \sqrt{{{\rm e}^{i \left ( bx+a \right ) }}+1},1/2\,\sqrt{2} \right ) \right ) }{\sqrt{i \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{3}-i{{\rm e}^{i \left ( bx+a \right ) }}}}} \right ) \sqrt{{\frac{i{{\rm e}^{i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*(b*x+2*I)*(exp(I*(b*x+a))^2-1)/b^2*2^(1/2)*(I*exp(I*(b*x+a))/(exp(I*(b*x+a))^2-1))^(1/2)/exp(I*(b*x+a))-2/b
^2*(-2*I*(-I+I*exp(I*(b*x+a))^2)/(exp(I*(b*x+a))*(-I+I*exp(I*(b*x+a))^2))^(1/2)-(exp(I*(b*x+a))+1)^(1/2)*(-2*e
xp(I*(b*x+a))+2)^(1/2)*(-exp(I*(b*x+a)))^(1/2)/(I*exp(I*(b*x+a))^3-I*exp(I*(b*x+a)))^(1/2)*(-2*EllipticE((exp(
I*(b*x+a))+1)^(1/2),1/2*2^(1/2))+EllipticF((exp(I*(b*x+a))+1)^(1/2),1/2*2^(1/2))))*2^(1/2)*(I*exp(I*(b*x+a))/(
exp(I*(b*x+a))^2-1))^(1/2)*(I*(exp(I*(b*x+a))^2-1)*exp(I*(b*x+a)))^(1/2)/exp(I*(b*x+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sqrt{\csc \left (b x + a\right )}\,{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)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)*sqrt(csc(b*x + a)), 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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos{\left (a + b x \right )} \sqrt{\csc{\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)**(1/2),x)

[Out]

Integral(x*cos(a + b*x)*sqrt(csc(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sqrt{\csc \left (b x + a\right )}\,{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)^(1/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)*sqrt(csc(b*x + a)), x)