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

Optimal. Leaf size=33 \[ \frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2}-\frac{2 x \sqrt{\cos (a+b x)}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0262149, antiderivative size = 33, 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 = {3444, 2639} \[ \frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2}-\frac{2 x \sqrt{\cos (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3444

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> -Simp[(x^(m - n
+ 1)*Cos[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cos[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 \sin (a+b x)}{\sqrt{\cos (a+b x)}} \, dx &=-\frac{2 x \sqrt{\cos (a+b x)}}{b}+\frac{2 \int \sqrt{\cos (a+b x)} \, dx}{b}\\ &=-\frac{2 x \sqrt{\cos (a+b x)}}{b}+\frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2}\\ \end{align*}

Mathematica [B]  time = 1.7706, size = 181, normalized size = 5.48 \[ \frac{4 \cos ^2\left (\frac{1}{2} (a+b x)\right )^{3/2} \sqrt{\frac{\cos (a+b x)}{(\cos (a+b x)+1)^2}} \sqrt{\frac{1}{\cos (a+b x)+1}} \left (-2 \sqrt{\sec ^2\left (\frac{1}{2} (a+b x)\right )} \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (a+b x)\right )\right ),-1\right )+\left (2 \tan \left (\frac{1}{2} (a+b x)\right )-b x\right ) \sqrt{\cos (a+b x) \sec ^2\left (\frac{1}{2} (a+b x)\right )}+2 \sqrt{\sec ^2\left (\frac{1}{2} (a+b x)\right )} E\left (\left .\sin ^{-1}\left (\tan \left (\frac{1}{2} (a+b x)\right )\right )\right |-1\right )\right )}{b^2 \sqrt{\frac{\cos (a+b x)}{\cos (a+b x)+1}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*(Cos[(a + b*x)/2]^2)^(3/2)*Sqrt[Cos[a + b*x]/(1 + Cos[a + b*x])^2]*Sqrt[(1 + Cos[a + b*x])^(-1)]*(2*Ellipti
cE[ArcSin[Tan[(a + b*x)/2]], -1]*Sqrt[Sec[(a + b*x)/2]^2] - 2*EllipticF[ArcSin[Tan[(a + b*x)/2]], -1]*Sqrt[Sec
[(a + b*x)/2]^2] + Sqrt[Cos[a + b*x]*Sec[(a + b*x)/2]^2]*(-(b*x) + 2*Tan[(a + b*x)/2])))/(b^2*Sqrt[Cos[a + b*x
]/(1 + Cos[a + b*x])])

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Maple [C]  time = 0.212, size = 310, normalized size = 9.4 \begin{align*} -{\frac{ \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{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}+1}{{{\rm e}^{i \left ( bx+a \right ) }}}}}}}}-{\frac{2\,i\sqrt{2}}{{b}^{2}{{\rm e}^{i \left ( bx+a \right ) }}} \left ( -2\,{\frac{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}+1}{\sqrt{ \left ( \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}+1 \right ){{\rm e}^{i \left ( bx+a \right ) }}}}}+{i\sqrt{2}\sqrt{-i \left ({{\rm e}^{i \left ( bx+a \right ) }}+i \right ) }\sqrt{i \left ({{\rm e}^{i \left ( bx+a \right ) }}-i \right ) }\sqrt{i{{\rm e}^{i \left ( bx+a \right ) }}} \left ( -2\,i{\it EllipticE} \left ( \sqrt{-i \left ({{\rm e}^{i \left ( bx+a \right ) }}+i \right ) },{\frac{\sqrt{2}}{2}} \right ) +i{\it EllipticF} \left ( \sqrt{-i \left ({{\rm e}^{i \left ( bx+a \right ) }}+i \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{3}+{{\rm e}^{i \left ( bx+a \right ) }}}}}} \right ) \sqrt{ \left ( \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}+1 \right ){{\rm e}^{i \left ( bx+a \right ) }}}{\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}+1}{{{\rm e}^{i \left ( bx+a \right ) }}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+2*I)*(exp(I*(b*x+a))^2+1)/b^2*2^(1/2)/((exp(I*(b*x+a))^2+1)/exp(I*(b*x+a)))^(1/2)/exp(I*(b*x+a))-2*I/b^2
*(-2*(exp(I*(b*x+a))^2+1)/((exp(I*(b*x+a))^2+1)*exp(I*(b*x+a)))^(1/2)+I*(-I*(exp(I*(b*x+a))+I))^(1/2)*2^(1/2)*
(I*(exp(I*(b*x+a))-I))^(1/2)*(I*exp(I*(b*x+a)))^(1/2)/(exp(I*(b*x+a))^3+exp(I*(b*x+a)))^(1/2)*(-2*I*EllipticE(
(-I*(exp(I*(b*x+a))+I))^(1/2),1/2*2^(1/2))+I*EllipticF((-I*(exp(I*(b*x+a))+I))^(1/2),1/2*2^(1/2))))*2^(1/2)/((
exp(I*(b*x+a))^2+1)/exp(I*(b*x+a)))^(1/2)*((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 \sin \left (b x + a\right )}{\sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*sin(b*x + a)/sqrt(cos(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*sin(b*x+a)/cos(b*x+a)^(1/2),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*sin(b*x+a)/cos(b*x+a)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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