\(\int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx\) [16]

Optimal result

Integrand size = 10, antiderivative size = 16 \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=-\frac {\cos (x) \sin (x)}{\sqrt {a \sin ^4(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, 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, 3852, 8} \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=-\frac {\sin (x) \cos (x)}{\sqrt {a \sin ^4(x)}} \]

[In]

Int[1/Sqrt[a*Sin[x]^4],x]

[Out]

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

Rule 8

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

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
]])

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^2(x) \int \csc ^2(x) \, dx}{\sqrt {a \sin ^4(x)}} \\ & = -\frac {\sin ^2(x) \text {Subst}(\int 1 \, dx,x,\cot (x))}{\sqrt {a \sin ^4(x)}} \\ & = -\frac {\cos (x) \sin (x)}{\sqrt {a \sin ^4(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=-\frac {\cos (x) \sin (x)}{\sqrt {a \sin ^4(x)}} \]

[In]

Integrate[1/Sqrt[a*Sin[x]^4],x]

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {\cos \left (x \right ) \sin \left (x \right ) \sqrt {16}}{4 \sqrt {a \left (\sin ^{4}\left (x \right )\right )}}\)	\(18\)
risch	\(\frac {2 i \left (1-{\mathrm e}^{-2 i x}\right )}{\sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}\)	\(31\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=\frac {\sqrt {a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a} \cos \left (x\right )}{{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=\int \frac {1}{\sqrt {a \sin ^{4}{\left (x \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(a*sin(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=-\frac {1}{\sqrt {a} \tan \left (x\right )} \]

[In]

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

[Out]

-1/(sqrt(a)*tan(x))

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=-\frac {1}{\sqrt {a} \tan \left (x\right )} \]

[In]

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

[Out]

-1/(sqrt(a)*tan(x))

Mupad [B] (verification not implemented)

Time = 13.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {a \sin ^4(x)}} \, dx=-\frac {\mathrm {cot}\left (x\right )}{\sqrt {a}} \]

[In]

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

[Out]

-cot(x)/a^(1/2)