3.11 \(\int \sin ^{\frac{3}{2}}(b x) \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 F\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{3 b}-\frac{2 \sqrt{\sin (b x)} \cos (b x)}{3 b} \]

[Out]

(-2*EllipticF[Pi/4 - (b*x)/2, 2])/(3*b) - (2*Cos[b*x]*Sqrt[Sin[b*x]])/(3*b)

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Rubi [A]  time = 0.0154634, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2641} \[ -\frac{2 F\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{3 b}-\frac{2 \sqrt{\sin (b x)} \cos (b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*EllipticF[Pi/4 - (b*x)/2, 2])/(3*b) - (2*Cos[b*x]*Sqrt[Sin[b*x]])/(3*b)

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 2641

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

Rubi steps

\begin{align*} \int \sin ^{\frac{3}{2}}(b x) \, dx &=-\frac{2 \cos (b x) \sqrt{\sin (b x)}}{3 b}+\frac{1}{3} \int \frac{1}{\sqrt{\sin (b x)}} \, dx\\ &=-\frac{2 F\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{3 b}-\frac{2 \cos (b x) \sqrt{\sin (b x)}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0361527, size = 33, normalized size = 0.8 \[ -\frac{2 \left (F\left (\left .\frac{1}{4} (\pi -2 b x)\right |2\right )+\sqrt{\sin (b x)} \cos (b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 72, normalized size = 1.8 \begin{align*}{\frac{1}{b\cos \left ( bx \right ) } \left ({\frac{1}{3}\sqrt{\sin \left ( bx \right ) +1}\sqrt{-2\,\sin \left ( bx \right ) +2}\sqrt{-\sin \left ( bx \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }-{\frac{2\, \left ( \cos \left ( bx \right ) \right ) ^{2}\sin \left ( bx \right ) }{3}} \right ){\frac{1}{\sqrt{\sin \left ( bx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (b x\right )^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x)^(3/2),x, algorithm="fricas")

[Out]

integral(sin(b*x)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{\frac{3}{2}}{\left (b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x)**(3/2),x)

[Out]

Integral(sin(b*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sin(b*x)^(3/2), x)