3.27 \(\int (c \sin (a+b x))^{3/2} \, dx\)

Optimal. Leaf size=75 \[ \frac{2 c^2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b \sqrt{c \sin (a+b x)}}-\frac{2 c \cos (a+b x) \sqrt{c \sin (a+b x)}}{3 b} \]

[Out]

(2*c^2*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*b*Sqrt[c*Sin[a + b*x]]) - (2*c*Cos[a + b*x]*Sqr
t[c*Sin[a + b*x]])/(3*b)

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Rubi [A]  time = 0.0326279, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2642, 2641} \[ \frac{2 c^2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b \sqrt{c \sin (a+b x)}}-\frac{2 c \cos (a+b x) \sqrt{c \sin (a+b x)}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(3/2),x]

[Out]

(2*c^2*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*b*Sqrt[c*Sin[a + b*x]]) - (2*c*Cos[a + b*x]*Sqr
t[c*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 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [A]  time = 0.0516922, size = 62, normalized size = 0.83 \[ -\frac{2 (c \sin (a+b x))^{3/2} \left (F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )+\sqrt{\sin (a+b x)} \cos (a+b x)\right )}{3 b \sin ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.034, size = 97, normalized size = 1.3 \begin{align*} -{\frac{{c}^{2}}{3\,b\cos \left ( bx+a \right ) } \left ( \sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2}\sqrt{\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}+2\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*sin(b*x + a))^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c \sin \left (b x + a\right )} c \sin \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*sin(b*x + a))*c*sin(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*sin(a + b*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*sin(b*x + a))^(3/2), x)