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

Optimal. Leaf size=47 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \sqrt {\sin (a+b x)} \cos (a+b x)}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2635, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \sqrt {\sin (a+b x)} \cos (a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^(3/2),x]

[Out]

(2*EllipticF[(a - Pi/2 + b*x)/2, 2])/(3*b) - (2*Cos[a + b*x]*Sqrt[Sin[a + 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}}(a+b x) \, dx &=-\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x) \sqrt {\sin (a+b x)}}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.85 \[ -\frac {2 \left (F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+\sqrt {\sin (a+b x)} \cos (a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^(3/2),x]

[Out]

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

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sin \left (b x + a\right )^{\frac {3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin(b*x + a)^(3/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x + a)^(3/2), x)

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maple [A]  time = 0.04, size = 88, normalized size = 1.87 \[ \frac {\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x + a)^(3/2), x)

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mupad [B]  time = 0.44, size = 42, normalized size = 0.89 \[ -\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{5/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^(3/2),x)

[Out]

-(cos(a + b*x)*sin(a + b*x)^(5/2)*hypergeom([-1/4, 1/2], 3/2, cos(a + b*x)^2))/(b*(sin(a + b*x)^2)^(5/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x)**(3/2), x)

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