3.17 \(\int \frac{1}{(a \sin ^4(x))^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\sin (x) \cos (x)}{a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}} \]

[Out]

(-2*Cos[x]^2*Cot[x])/(3*a*Sqrt[a*Sin[x]^4]) - (Cos[x]^2*Cot[x]^3)/(5*a*Sqrt[a*Sin[x]^4]) - (Cos[x]*Sin[x])/(a*
Sqrt[a*Sin[x]^4])

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Rubi [A]  time = 0.0195453, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3767} \[ -\frac{\sin (x) \cos (x)}{a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[x]^2*Cot[x])/(3*a*Sqrt[a*Sin[x]^4]) - (Cos[x]^2*Cot[x]^3)/(5*a*Sqrt[a*Sin[x]^4]) - (Cos[x]*Sin[x])/(a*
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]

Rubi steps

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

Mathematica [A]  time = 0.0334892, size = 34, normalized size = 0.5 \[ -\frac{\sin ^5(x) \cos (x) \left (3 \csc ^4(x)+4 \csc ^2(x)+8\right )}{15 \left (a \sin ^4(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[x]*(8 + 4*Csc[x]^2 + 3*Csc[x]^4)*Sin[x]^5)/(15*(a*Sin[x]^4)^(3/2))

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Maple [A]  time = 0.119, size = 29, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( x \right ) \right ) ^{4}-20\, \left ( \cos \left ( x \right ) \right ) ^{2}+15 \right ) \sin \left ( x \right ) \cos \left ( x \right ) }{15} \left ( a \left ( \sin \left ( x \right ) \right ) ^{4} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sin(x)^4)^(3/2),x)

[Out]

-1/15*(8*cos(x)^4-20*cos(x)^2+15)*sin(x)*cos(x)/(a*sin(x)^4)^(3/2)

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Maxima [A]  time = 1.44756, size = 31, normalized size = 0.46 \begin{align*} -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, a^{\frac{3}{2}} \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)^4)^(3/2),x, algorithm="maxima")

[Out]

-1/15*(15*tan(x)^4 + 10*tan(x)^2 + 3)/(a^(3/2)*tan(x)^5)

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Fricas [A]  time = 1.62095, size = 196, normalized size = 2.88 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}{\left (8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )}}{15 \,{\left (a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)^4)^(3/2),x, algorithm="fricas")

[Out]

1/15*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(8*cos(x)^5 - 20*cos(x)^3 + 15*cos(x))/((a^2*cos(x)^6 - 3*a^2*cos(x)^
4 + 3*a^2*cos(x)^2 - a^2)*sin(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin ^{4}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)**4)**(3/2),x)

[Out]

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

[Out]

Exception raised: TypeError