3.15 \(\int \sqrt{a \sin ^4(x)} \, dx\)

Optimal. Leaf size=36 \[ \frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)} \]

[Out]

-(Cot[x]*Sqrt[a*Sin[x]^4])/2 + (x*Csc[x]^2*Sqrt[a*Sin[x]^4])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0131223, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 8} \[ \frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Sin[x]^4],x]

[Out]

-(Cot[x]*Sqrt[a*Sin[x]^4])/2 + (x*Csc[x]^2*Sqrt[a*Sin[x]^4])/2

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 8

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

Rubi steps

\begin{align*} \int \sqrt{a \sin ^4(x)} \, dx &=\left (\csc ^2(x) \sqrt{a \sin ^4(x)}\right ) \int \sin ^2(x) \, dx\\ &=-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}+\frac{1}{2} \left (\csc ^2(x) \sqrt{a \sin ^4(x)}\right ) \int 1 \, dx\\ &=-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}+\frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}\\ \end{align*}

Mathematica [A]  time = 0.0164401, size = 25, normalized size = 0.69 \[ \frac{1}{2} \csc (x) \sqrt{a \sin ^4(x)} (x \csc (x)-\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Sin[x]^4],x]

[Out]

(Csc[x]*(-Cos[x] + x*Csc[x])*Sqrt[a*Sin[x]^4])/2

________________________________________________________________________________________

Maple [A]  time = 0.158, size = 27, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{16} \left ( \sin \left ( x \right ) \cos \left ( x \right ) -x \right ) }{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(x)^4)^(1/2),x)

[Out]

-1/8*16^(1/2)*(a*sin(x)^4)^(1/2)*(sin(x)*cos(x)-x)/sin(x)^2

________________________________________________________________________________________

Maxima [A]  time = 1.44084, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{2} \, \sqrt{a} x - \frac{\sqrt{a} \tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^4)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(a)*x - 1/2*sqrt(a)*tan(x)/(tan(x)^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.6705, size = 103, normalized size = 2.86 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}{\left (\cos \left (x\right ) \sin \left (x\right ) - x\right )}}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^4)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(cos(x)*sin(x) - x)/(cos(x)^2 - 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin ^{4}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)**4)**(1/2),x)

[Out]

Integral(sqrt(a*sin(x)**4), x)

________________________________________________________________________________________

Giac [A]  time = 1.09986, size = 20, normalized size = 0.56 \begin{align*} \frac{1}{4} \, \sqrt{a}{\left (2 \, x - \sin \left (2 \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(x)^4)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(a)*(2*x - sin(2*x))