\(\int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx\) [330]

Optimal result

Integrand size = 18, antiderivative size = 60 \[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 x \cos ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{9 b^2}+\frac {4 \sqrt {\cos (a+b x)} \sin (a+b x)}{9 b^2} \]

[Out]

-2/3*x*cos(b*x+a)^(3/2)/b+4/9*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(sin(1/2*a+1/2*b*x),2^(
1/2))/b^2+4/9*sin(b*x+a)*cos(b*x+a)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3525, 2715, 2720} \[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{9 b^2}+\frac {4 \sin (a+b x) \sqrt {\cos (a+b x)}}{9 b^2}-\frac {2 x \cos ^{\frac {3}{2}}(a+b x)}{3 b} \]

[In]

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

[Out]

(-2*x*Cos[a + b*x]^(3/2))/(3*b) + (4*EllipticF[(a + b*x)/2, 2])/(9*b^2) + (4*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/
(9*b^2)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3525

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*} \text {integral}& = -\frac {2 x \cos ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \cos ^{\frac {3}{2}}(a+b x) \, dx}{3 b} \\ & = -\frac {2 x \cos ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \sqrt {\cos (a+b x)} \sin (a+b x)}{9 b^2}+\frac {2 \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{9 b} \\ & = -\frac {2 x \cos ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{9 b^2}+\frac {4 \sqrt {\cos (a+b x)} \sin (a+b x)}{9 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+2 \sqrt {\cos (a+b x)} (-3 b x \cos (a+b x)+2 \sin (a+b x))}{9 b^2} \]

[In]

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

[Out]

(4*EllipticF[(a + b*x)/2, 2] + 2*Sqrt[Cos[a + b*x]]*(-3*b*x*Cos[a + b*x] + 2*Sin[a + b*x]))/(9*b^2)

Maple [F]

\[\int x \sin \left (x b +a \right ) \sqrt {\cos \left (x b +a \right )}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\int x \sin {\left (a + b x \right )} \sqrt {\cos {\left (a + b x \right )}}\, dx \]

[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)

Maxima [F]

\[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\int { x \sqrt {\cos \left (b x + a\right )} \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\int { x \sqrt {\cos \left (b x + a\right )} \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {\cos (a+b x)} \sin (a+b x) \, dx=\int x\,\sqrt {\cos \left (a+b\,x\right )}\,\sin \left (a+b\,x\right ) \,d x \]

[In]

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

[Out]

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