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]

4*(cos(1/2*b*x+1/2*a)^2)^(1/2)/cos(1/2*b*x+1/2*a)*EllipticE(sin(1/2*b*x+1/2*a),2^(1/2))/b^2-2*x*cos(b*x+a)^(1/
2)/b

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [B]  time = 1.76, size = 181, normalized size = 5.48 \[ \frac {4 \cos ^2\left (\frac {1}{2} (a+b x)\right )^{3/2} \sqrt {\frac {\cos (a+b x)}{(\cos (a+b x)+1)^2}} \sqrt {\frac {1}{\cos (a+b x)+1}} \left (\left (2 \tan \left (\frac {1}{2} (a+b x)\right )-b x\right ) \sqrt {\cos (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )}-2 \sqrt {\sec ^2\left (\frac {1}{2} (a+b x)\right )} F\left (\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (a+b x)\right )\right )\right |-1\right )+2 \sqrt {\sec ^2\left (\frac {1}{2} (a+b x)\right )} E\left (\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (a+b x)\right )\right )\right |-1\right )\right )}{b^2 \sqrt {\frac {\cos (a+b x)}{\cos (a+b x)+1}}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\sqrt {\cos \left (b x + a\right )}}\,{d x} \]

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)

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maple [C]  time = 0.16, size = 310, normalized size = 9.39 \[ -\frac {\left (b x +2 i\right ) \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) \sqrt {2}\, {\mathrm e}^{-i \left (b x +a \right )}}{b^{2} \sqrt {\left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) {\mathrm e}^{-i \left (b x +a \right )}}}-\frac {2 i \left (-\frac {2 \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{\sqrt {\left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) {\mathrm e}^{i \left (b x +a \right )}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 i \left (b x +a \right )}+{\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) {\mathrm e}^{i \left (b x +a \right )}}\, {\mathrm e}^{-i \left (b x +a \right )}}{b^{2} \sqrt {\left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) {\mathrm e}^{-i \left (b x +a \right )}}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\sqrt {\cos \left (b x + a\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\sin \left (a+b\,x\right )}{\sqrt {\cos \left (a+b\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin {\left (a + b x \right )}}{\sqrt {\cos {\left (a + b x \right )}}}\, dx \]

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]

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

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