3.345 \(\int x \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x) \, dx\)

Optimal. Leaf size=65 \[ -\frac{12 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{4 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b} \]

[Out]

(-12*EllipticE[(a - Pi/2 + b*x)/2, 2])/(25*b^2) + (4*Cos[a + b*x]*Sin[a + b*x]^(3/2))/(25*b^2) + (2*x*Sin[a +
b*x]^(5/2))/(5*b)

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Rubi [A]  time = 0.0323581, antiderivative size = 65, 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 = {3443, 2635, 2639} \[ -\frac{12 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{4 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*x]*Sin[a + b*x]^(3/2),x]

[Out]

(-12*EllipticE[(a - Pi/2 + b*x)/2, 2])/(25*b^2) + (4*Cos[a + b*x]*Sin[a + b*x]^(3/2))/(25*b^2) + (2*x*Sin[a +
b*x]^(5/2))/(5*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 2635

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] && Integer
Q[2*n]

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 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{2 \int \sin ^{\frac{5}{2}}(a+b x) \, dx}{5 b}\\ &=\frac{4 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{6 \int \sqrt{\sin (a+b x)} \, dx}{25 b}\\ &=-\frac{12 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{25 b^2}+\frac{4 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}\\ \end{align*}

Mathematica [C]  time = 0.925856, size = 108, normalized size = 1.66 \[ \frac{\sqrt{\sin (a+b x)} \left (4 \tan \left (\frac{1}{2} (a+b x)\right ) \sqrt{\sec ^2\left (\frac{1}{2} (a+b x)\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )+2 \sin (2 (a+b x))-5 b x \cos (2 (a+b x))-12 \tan \left (\frac{1}{2} (a+b x)\right )+5 b x\right )}{25 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]*Sin[a + b*x]^(3/2),x]

[Out]

(Sqrt[Sin[a + b*x]]*(5*b*x - 5*b*x*Cos[2*(a + b*x)] + 2*Sin[2*(a + b*x)] - 12*Tan[(a + b*x)/2] + 4*Hypergeomet
ric2F1[1/2, 3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sqrt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(25*b^2)

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(b*x+a)*sin(b*x+a)^(3/2),x)

[Out]

int(x*cos(b*x+a)*sin(b*x+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac{3}{2}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)*sin(b*x + a)^(3/2), 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)^(3/2),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*cos(b*x+a)*sin(b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac{3}{2}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)*sin(b*x + a)^(3/2), x)