3.347 \(\int \frac{x \cos (a+b x)}{\sqrt{\sin (a+b x)}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 x \sqrt{\sin (a+b x)}}{b}-\frac{4 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0222333, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3443, 2639} \[ \frac{2 x \sqrt{\sin (a+b x)}}{b}-\frac{4 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cos[a + b*x])/Sqrt[Sin[a + b*x]],x]

[Out]

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

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 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)}{\sqrt{\sin (a+b x)}} \, dx &=\frac{2 x \sqrt{\sin (a+b x)}}{b}-\frac{2 \int \sqrt{\sin (a+b x)} \, dx}{b}\\ &=-\frac{4 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{b^2}+\frac{2 x \sqrt{\sin (a+b x)}}{b}\\ \end{align*}

Mathematica [C]  time = 1.18553, size = 86, normalized size = 2.26 \[ \frac{2 \sqrt{\sin (a+b x)} \left (2 \tan \left (\frac{1}{2} (a+b x)\right ) \sqrt{\sec ^2\left (\frac{1}{2} (a+b x)\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )-6 \tan \left (\frac{1}{2} (a+b x)\right )+3 b x\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Cos[a + b*x])/Sqrt[Sin[a + b*x]],x]

[Out]

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

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Maple [C]  time = 0.204, size = 308, normalized size = 8.1 \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 ) }}}{\frac{1}{\sqrt{{\frac{-i \left ( \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}-1 \right ) }{{{\rm e}^{i \left ( bx+a \right ) }}}}}}}}-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 ){\frac{1}{\sqrt{{\frac{-i \left ( \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}-1 \right ) }{{{\rm e}^{i \left ( bx+a \right ) }}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(b*x+a)/sin(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))^2-1)/exp(I*(b*x+a)))^(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*exp
(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))^
2-1)/exp(I*(b*x+a)))^(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 \frac{x \cos \left (b x + a\right )}{\sqrt{\sin \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)/sin(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)/sqrt(sin(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)/sin(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 \frac{x \cos{\left (a + b x \right )}}{\sqrt{\sin{\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)/sin(b*x+a)**(1/2),x)

[Out]

Integral(x*cos(a + b*x)/sqrt(sin(a + b*x)), 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 )}{\sqrt{\sin \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)/sin(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)/sqrt(sin(b*x + a)), x)