3.1.14 \(\int (a \sin ^4(x))^{3/2} \, dx\) [14]

Optimal. Leaf size=78 \[ -\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)} \]

[Out]

-5/16*a*cot(x)*(a*sin(x)^4)^(1/2)+5/16*a*x*csc(x)^2*(a*sin(x)^4)^(1/2)-5/24*a*cos(x)*sin(x)*(a*sin(x)^4)^(1/2)
-1/6*a*cos(x)*sin(x)^3*(a*sin(x)^4)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 8} \begin {gather*} -\frac {5}{24} a \sin (x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \sin ^3(x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sin[x]^4)^(3/2),x]

[Out]

(-5*a*Cot[x]*Sqrt[a*Sin[x]^4])/16 + (5*a*x*Csc[x]^2*Sqrt[a*Sin[x]^4])/16 - (5*a*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^4]
)/24 - (a*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^4])/6

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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] && Integ
erQ[2*n]

Rule 3286

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 ^4(x)\right )^{3/2} \, dx &=\left (a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^6(x) \, dx\\ &=-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}+\frac {1}{6} \left (5 a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^4(x) \, dx\\ &=-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}+\frac {1}{8} \left (5 a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^2(x) \, dx\\ &=-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}+\frac {1}{16} \left (5 a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int 1 \, dx\\ &=-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 38, normalized size = 0.49 \begin {gather*} -\frac {1}{192} \csc ^6(x) \left (a \sin ^4(x)\right )^{3/2} (-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[x]^4)^(3/2),x]

[Out]

-1/192*(Csc[x]^6*(a*Sin[x]^4)^(3/2)*(-60*x + 45*Sin[2*x] - 9*Sin[4*x] + Sin[6*x]))

________________________________________________________________________________________

Maple [A]
time = 0.23, size = 47, normalized size = 0.60

method	result	size
default	\(-\frac {\left (a \left (1-\left (\cos ^{2}\left (x \right )\right )\right )^{2}\right )^{\frac {3}{2}} \left (8 \left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )-26 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )+33 \sin \left (x \right ) \cos \left (x \right )-15 x \right )}{48 \sin \left (x \right )^{6}}\)	\(47\)
risch	\(-\frac {5 a \,{\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}\, x}{16 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {i a \,{\mathrm e}^{8 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{384 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {3 i a \,{\mathrm e}^{6 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {15 i a \,{\mathrm e}^{4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {15 i a \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {3 i a \,{\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {i a \,{\mathrm e}^{-4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{384 \left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(249\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(x)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/48*(a*(1-cos(x)^2)^2)^(3/2)*(8*cos(x)^5*sin(x)-26*cos(x)^3*sin(x)+33*sin(x)*cos(x)-15*x)/sin(x)^6

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 55, normalized size = 0.71 \begin {gather*} \frac {5}{16} \, a^{\frac {3}{2}} x - \frac {33 \, a^{\frac {3}{2}} \tan \left (x\right )^{5} + 40 \, a^{\frac {3}{2}} \tan \left (x\right )^{3} + 15 \, a^{\frac {3}{2}} \tan \left (x\right )}{48 \, {\left (\tan \left (x\right )^{6} + 3 \, \tan \left (x\right )^{4} + 3 \, \tan \left (x\right )^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^4)^(3/2),x, algorithm="maxima")

[Out]

5/16*a^(3/2)*x - 1/48*(33*a^(3/2)*tan(x)^5 + 40*a^(3/2)*tan(x)^3 + 15*a^(3/2)*tan(x))/(tan(x)^6 + 3*tan(x)^4 +
3*tan(x)^2 + 1)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 56, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a} {\left (15 \, a x - {\left (8 \, a \cos \left (x\right )^{5} - 26 \, a \cos \left (x\right )^{3} + 33 \, a \cos \left (x\right )\right )} \sin \left (x\right )\right )}}{48 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^4)^(3/2),x, algorithm="fricas")

[Out]

-1/48*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(15*a*x - (8*a*cos(x)^5 - 26*a*cos(x)^3 + 33*a*cos(x))*sin(x))/(cos(
x)^2 - 1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sin ^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)**4)**(3/2),x)

[Out]

Integral((a*sin(x)**4)**(3/2), x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 27, normalized size = 0.35 \begin {gather*} \frac {1}{192} \, a^{\frac {3}{2}} {\left (60 \, x - \sin \left (6 \, x\right ) + 9 \, \sin \left (4 \, x\right ) - 45 \, \sin \left (2 \, x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^4)^(3/2),x, algorithm="giac")

[Out]

1/192*a^(3/2)*(60*x - sin(6*x) + 9*sin(4*x) - 45*sin(2*x))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a\,{\sin \left (x\right )}^4\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(x)^4)^(3/2),x)

[Out]

int((a*sin(x)^4)^(3/2), x)

________________________________________________________________________________________