3.340 \(\int x \sqrt{\sec (a+b x)} \sin (a+b x) \, dx\)

Optimal. Leaf size=53 \[ \frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2}-\frac{2 x}{b \sqrt{\sec (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0366521, antiderivative size = 53, 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 = {4212, 3771, 2639} \[ \frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b^2}-\frac{2 x}{b \sqrt{\sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[Sec[a + b*x]]*Sin[a + b*x],x]

[Out]

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

Rule 4212

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

Mathematica [B]  time = 2.21082, size = 132, normalized size = 2.49 \[ \frac{2 \left (-\frac{2 \sec ^2\left (\frac{1}{2} (a+b x)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (a+b x)\right )\right ),-1\right )}{\sqrt{\cos (a+b x) \sec ^4\left (\frac{1}{2} (a+b x)\right )}}+2 \tan \left (\frac{1}{2} (a+b x)\right )+\frac{2 \sec ^2\left (\frac{1}{2} (a+b x)\right ) E\left (\left .\sin ^{-1}\left (\tan \left (\frac{1}{2} (a+b x)\right )\right )\right |-1\right )}{\sqrt{\cos (a+b x) \sec ^4\left (\frac{1}{2} (a+b x)\right )}}-b x\right )}{b^2 \sqrt{\sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[Sec[a + b*x]]*Sin[a + b*x],x]

[Out]

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

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Maple [C]  time = 0.184, size = 310, normalized size = 5.9 \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 ) }}}\sqrt{{\frac{{{\rm e}^{i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) ^{2}+1}}}}-{\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{{\frac{{{\rm e}^{i \left ( bx+a \right ) }}}{ \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 ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*sqrt(sec(b*x + a))*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*sin(b*x+a)*sec(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 \sin{\left (a + b x \right )} \sqrt{\sec{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)*sec(b*x+a)**(1/2),x)

[Out]

Integral(x*sin(a + b*x)*sqrt(sec(a + b*x)), x)

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

[Out]

integrate(x*sqrt(sec(b*x + a))*sin(b*x + a), x)