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

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

[Out]

-14/45*a*cos(x)*(a*sin(x)^3)^(1/2)-14/15*a*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/4*Pi+1/2*x)*EllipticE(cos(1/4*Pi+
1/2*x),2^(1/2))*(a*sin(x)^3)^(1/2)/sin(x)^(3/2)-2/9*a*cos(x)*sin(x)^2*(a*sin(x)^3)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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
]])

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*}

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Mathematica [A]  time = 0.10, 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))

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a \cos \relax (x)^{2} - a\right )} \sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)} \sin \relax (x), x\right ) \]

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)

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

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)

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maple [C]  time = 0.51, size = 337, normalized size = 4.62 \[ \frac {\left (21 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \cos \relax (x )-42 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \cos \relax (x )+21 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-42 \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticE \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-10 \left (\cos ^{5}\relax (x )\right )+34 \left (\cos ^{3}\relax (x )\right )-66 \cos \relax (x )+42\right ) \left (a \left (\sin ^{3}\relax (x )\right )\right )^{\frac {3}{2}}}{45 \sin \relax (x )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(21*((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))*2^(1/2)*cos(x)-42*((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)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2
),1/2*2^(1/2))*2^(1/2)*cos(x)+21*((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))*2^(1/2)-42*((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)*EllipticE(((I*cos(x)+sin(x
)-I)/sin(x))^(1/2),1/2*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

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

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\sin \relax (x)}^3\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*sin(x)^3)^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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