3.8 \(\int (a \sin ^3(x))^{3/2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2}{9} a \sin ^2(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{14 a E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{15 \sin ^{\frac{3}{2}}(x)} \]

[Out]

(-14*a*Cos[x]*Sqrt[a*Sin[x]^3])/45 - (14*a*EllipticE[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(15*Sin[x]^(3/2)) - (2*a
*Cos[x]*Sin[x]^2*Sqrt[a*Sin[x]^3])/9

________________________________________________________________________________________

Rubi [A]  time = 0.0247842, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2639} \[ -\frac{2}{9} a \sin ^2(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{14 a E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{15 \sin ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14*a*Cos[x]*Sqrt[a*Sin[x]^3])/45 - (14*a*EllipticE[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(15*Sin[x]^(3/2)) - (2*a
*Cos[x]*Sin[x]^2*Sqrt[a*Sin[x]^3])/9

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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 \left (a \sin ^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \sin ^3(x)}\right ) \int \sin ^{\frac{9}{2}}(x) \, dx}{\sin ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{9} a \cos (x) \sin ^2(x) \sqrt{a \sin ^3(x)}+\frac{\left (7 a \sqrt{a \sin ^3(x)}\right ) \int \sin ^{\frac{5}{2}}(x) \, dx}{9 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{2}{9} a \cos (x) \sin ^2(x) \sqrt{a \sin ^3(x)}+\frac{\left (7 a \sqrt{a \sin ^3(x)}\right ) \int \sqrt{\sin (x)} \, dx}{15 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{14 a E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{15 \sin ^{\frac{3}{2}}(x)}-\frac{2}{9} a \cos (x) \sin ^2(x) \sqrt{a \sin ^3(x)}\\ \end{align*}

Mathematica [A]  time = 0.100251, size = 54, normalized size = 0.74 \[ \frac{\left (a \sin ^3(x)\right )^{3/2} \left (\sqrt{\sin (x)} (5 \sin (4 x)-38 \sin (2 x))-168 E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )\right )}{180 \sin ^{\frac{9}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*Sin[x]^3)^(3/2)*(-168*EllipticE[(Pi - 2*x)/4, 2] + Sqrt[Sin[x]]*(-38*Sin[2*x] + 5*Sin[4*x])))/(180*Sin[x]^
(9/2))

________________________________________________________________________________________

Maple [C]  time = 0.213, size = 337, normalized size = 4.6 \begin{align*} -{\frac{1}{45\, \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( 42\,\sqrt{2}\cos \left ( x \right ) \sqrt{{\frac{-i\cos \left ( x \right ) +\sin \left ( x \right ) +i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}-21\,\sqrt{2}\cos \left ( x \right ) \sqrt{{\frac{-i\cos \left ( x \right ) +\sin \left ( x \right ) +i}{\sin \left ( x \right ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) +42\,\sqrt{2}\sqrt{{\frac{-i\cos \left ( x \right ) +\sin \left ( x \right ) +i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}-21\,\sqrt{2}\sqrt{{\frac{-i\cos \left ( x \right ) +\sin \left ( x \right ) +i}{\sin \left ( x \right ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) +10\, \left ( \cos \left ( x \right ) \right ) ^{5}-34\, \left ( \cos \left ( x \right ) \right ) ^{3}+66\,\cos \left ( x \right ) -42 \right ) \left ( a \left ( \sin \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45*(42*2^(1/2)*cos(x)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*
2^(1/2))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)-21*2^(1/2)*cos(x)*((-I*cos(x)+sin(x)
+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)
-I)/sin(x))^(1/2),1/2*2^(1/2))+42*2^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/s
in(x))^(1/2),1/2*2^(1/2))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)-21*2^(1/2)*((-I*cos
(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*cos
(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))+10*cos(x)^5-34*cos(x)^3+66*cos(x)-42)*(a*sin(x)^3)^(3/2)/sin(x)^5

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(x)^3)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*cos(x)^2 - a)*sqrt(-(a*cos(x)^2 - a)*sin(x))*sin(x), x)

________________________________________________________________________________________

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((a*sin(x)**3)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(x)^3)^(3/2), x)