\(\int \sqrt {a \sin ^4(x)} \, dx\) [15]

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \sqrt {a \sin ^4(x)} \, dx=-\frac {1}{2} \cot (x) \sqrt {a \sin ^4(x)}+\frac {1}{2} x \csc ^2(x) \sqrt {a \sin ^4(x)} \]

[Out]

-1/2*cot(x)*(a*sin(x)^4)^(1/2)+1/2*x*csc(x)^2*(a*sin(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 8} \[ \int \sqrt {a \sin ^4(x)} \, dx=\frac {1}{2} x \csc ^2(x) \sqrt {a \sin ^4(x)}-\frac {1}{2} \cot (x) \sqrt {a \sin ^4(x)} \]

[In]

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

[Out]

-1/2*(Cot[x]*Sqrt[a*Sin[x]^4]) + (x*Csc[x]^2*Sqrt[a*Sin[x]^4])/2

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \sqrt {a \sin ^4(x)} \, dx=\frac {1}{2} \csc (x) (-\cos (x)+x \csc (x)) \sqrt {a \sin ^4(x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {\sqrt {a \left (\sin ^{4}\left (x \right )\right )}\, \left (\cot \left (x \right )-\left (\csc ^{2}\left (x \right )\right ) x \right ) \sqrt {16}}{8}\)	\(24\)
risch	\(-\frac {\sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}\, {\mathrm e}^{2 i x} x}{2 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}\, {\mathrm e}^{4 i x}}{8 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(102\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \sin ^4(x)} \, dx=\frac {\sqrt {a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a} {\left (\cos \left (x\right ) \sin \left (x\right ) - x\right )}}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]

[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]

\[ \int \sqrt {a \sin ^4(x)} \, dx=\int \sqrt {a \sin ^{4}{\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a*sin(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \sqrt {a \sin ^4(x)} \, dx=\frac {1}{2} \, \sqrt {a} x - \frac {\sqrt {a} \tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.42 \[ \int \sqrt {a \sin ^4(x)} \, dx=\frac {1}{4} \, \sqrt {a} {\left (2 \, x - \sin \left (2 \, x\right )\right )} \]

[In]

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

[Out]

1/4*sqrt(a)*(2*x - sin(2*x))

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a \sin ^4(x)} \, dx=\int \sqrt {a\,{\sin \left (x\right )}^4} \,d x \]

[In]

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

[Out]

int((a*sin(x)^4)^(1/2), x)