3.7 \(\int (a \sin ^3(x))^{5/2} \, dx\)

Optimal. Leaf size=123 \[ -\frac{2}{15} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{26 a^2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{77 \sin ^{\frac{3}{2}}(x)} \]

[Out]

(-26*a^2*Cot[x]*Sqrt[a*Sin[x]^3])/77 - (26*a^2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(77*Sin[x]^(3/2)) -
(78*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^3])/385 - (26*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^3])/165 - (2*a^2*Cos[x]*Si
n[x]^5*Sqrt[a*Sin[x]^3])/15

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Rubi [A]  time = 0.0407798, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ -\frac{2}{15} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{26 a^2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{77 \sin ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sin[x]^3)^(5/2),x]

[Out]

(-26*a^2*Cot[x]*Sqrt[a*Sin[x]^3])/77 - (26*a^2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(77*Sin[x]^(3/2)) -
(78*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^3])/385 - (26*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^3])/165 - (2*a^2*Cos[x]*Si
n[x]^5*Sqrt[a*Sin[x]^3])/15

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 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 \left (a \sin ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \sin ^3(x)}\right ) \int \sin ^{\frac{15}{2}}(x) \, dx}{\sin ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{15} a^2 \cos (x) \sin ^5(x) \sqrt{a \sin ^3(x)}+\frac{\left (13 a^2 \sqrt{a \sin ^3(x)}\right ) \int \sin ^{\frac{11}{2}}(x) \, dx}{15 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{26}{165} a^2 \cos (x) \sin ^3(x) \sqrt{a \sin ^3(x)}-\frac{2}{15} a^2 \cos (x) \sin ^5(x) \sqrt{a \sin ^3(x)}+\frac{\left (39 a^2 \sqrt{a \sin ^3(x)}\right ) \int \sin ^{\frac{7}{2}}(x) \, dx}{55 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{78}{385} a^2 \cos (x) \sin (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \cos (x) \sin ^3(x) \sqrt{a \sin ^3(x)}-\frac{2}{15} a^2 \cos (x) \sin ^5(x) \sqrt{a \sin ^3(x)}+\frac{\left (39 a^2 \sqrt{a \sin ^3(x)}\right ) \int \sin ^{\frac{3}{2}}(x) \, dx}{77 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{78}{385} a^2 \cos (x) \sin (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \cos (x) \sin ^3(x) \sqrt{a \sin ^3(x)}-\frac{2}{15} a^2 \cos (x) \sin ^5(x) \sqrt{a \sin ^3(x)}+\frac{\left (13 a^2 \sqrt{a \sin ^3(x)}\right ) \int \frac{1}{\sqrt{\sin (x)}} \, dx}{77 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{26 a^2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{77 \sin ^{\frac{3}{2}}(x)}-\frac{78}{385} a^2 \cos (x) \sin (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \cos (x) \sin ^3(x) \sqrt{a \sin ^3(x)}-\frac{2}{15} a^2 \cos (x) \sin ^5(x) \sqrt{a \sin ^3(x)}\\ \end{align*}

Mathematica [A]  time = 0.188814, size = 65, normalized size = 0.53 \[ \frac{a \left (a \sin ^3(x)\right )^{3/2} \left (\sqrt{\sin (x)} (-15465 \cos (x)+3657 \cos (3 x)-749 \cos (5 x)+77 \cos (7 x))-12480 F\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )\right )}{36960 \sin ^{\frac{9}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[x]^3)^(5/2),x]

[Out]

(a*(-12480*EllipticF[(Pi - 2*x)/4, 2] + (-15465*Cos[x] + 3657*Cos[3*x] - 749*Cos[5*x] + 77*Cos[7*x])*Sqrt[Sin[
x]])*(a*Sin[x]^3)^(3/2))/(36960*Sin[x]^(9/2))

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Maple [C]  time = 0.224, size = 149, normalized size = 1.2 \begin{align*} -{\frac{1}{1155\, \left ( \sin \left ( x \right ) \right ) ^{7} \left ( -1+\cos \left ( x \right ) \right ) } \left ( -154\, \left ( \cos \left ( x \right ) \right ) ^{8}+195\,i\sqrt{2}\sin \left ( x \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \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 ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +154\, \left ( \cos \left ( x \right ) \right ) ^{7}+644\, \left ( \cos \left ( x \right ) \right ) ^{6}-644\, \left ( \cos \left ( x \right ) \right ) ^{5}-1060\, \left ( \cos \left ( x \right ) \right ) ^{4}+1060\, \left ( \cos \left ( x \right ) \right ) ^{3}+960\, \left ( \cos \left ( x \right ) \right ) ^{2}-960\,\cos \left ( x \right ) \right ) \left ( a \left ( \sin \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(x)^3)^(5/2),x)

[Out]

-1/1155*(-154*cos(x)^8+195*I*2^(1/2)*sin(x)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(
-(I*cos(x)-sin(x)-I)/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))+154*cos(x)^7+644*
cos(x)^6-644*cos(x)^5-1060*cos(x)^4+1060*cos(x)^3+960*cos(x)^2-960*cos(x))*(a*sin(x)^3)^(5/2)/sin(x)^7/(-1+cos
(x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sin(x)^3)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sqrt{-{\left (a \cos \left (x\right )^{2} - a\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)^(5/2),x, algorithm="fricas")

[Out]

integral(-(a^2*cos(x)^6 - 3*a^2*cos(x)^4 + 3*a^2*cos(x)^2 - a^2)*sqrt(-(a*cos(x)^2 - a)*sin(x)), x)

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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)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*sin(x)^3)^(5/2), x)