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

Optimal. Leaf size=16 \[ -\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0136734, antiderivative size = 16, 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, 3767, 8} \[ -\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3767

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]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \sin ^4(x)}} \, dx &=\frac{\sin ^2(x) \int \csc ^2(x) \, dx}{\sqrt{a \sin ^4(x)}}\\ &=-\frac{\sin ^2(x) \operatorname{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]  time = 0.0071261, size = 16, normalized size = 1. \[ -\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.159, size = 15, normalized size = 0.9 \begin{align*} -{\sin \left ( x \right ) \cos \left ( x \right ){\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.4513, size = 12, normalized size = 0.75 \begin{align*} -\frac{1}{\sqrt{a} \tan \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.54785, size = 95, normalized size = 5.94 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin ^{4}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*sin(x)**4), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError