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

Optimal. Leaf size=77 \[ -\frac{10 \cos (x)}{21 a \sqrt{a \sin ^3(x)}}-\frac{10 \sin ^{\frac{3}{2}}(x) F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{21 a \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}} \]

[Out]

(-10*Cos[x])/(21*a*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x])/(7*a*Sqrt[a*Sin[x]^3]) - (10*EllipticF[Pi/4 - x/2, 2]
*Sin[x]^(3/2))/(21*a*Sqrt[a*Sin[x]^3])

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Rubi [A]  time = 0.0254569, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2641} \[ -\frac{10 \cos (x)}{21 a \sqrt{a \sin ^3(x)}}-\frac{10 \sin ^{\frac{3}{2}}(x) F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{21 a \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Cos[x])/(21*a*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x])/(7*a*Sqrt[a*Sin[x]^3]) - (10*EllipticF[Pi/4 - x/2, 2]
*Sin[x]^(3/2))/(21*a*Sqrt[a*Sin[x]^3])

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 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx &=\frac{\sin ^{\frac{3}{2}}(x) \int \frac{1}{\sin ^{\frac{9}{2}}(x)} \, dx}{a \sqrt{a \sin ^3(x)}}\\ &=-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}}+\frac{\left (5 \sin ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sin ^{\frac{5}{2}}(x)} \, dx}{7 a \sqrt{a \sin ^3(x)}}\\ &=-\frac{10 \cos (x)}{21 a \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}}+\frac{\left (5 \sin ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\sin (x)}} \, dx}{21 a \sqrt{a \sin ^3(x)}}\\ &=-\frac{10 \cos (x)}{21 a \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}}-\frac{10 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sin ^{\frac{3}{2}}(x)}{21 a \sqrt{a \sin ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0678059, size = 48, normalized size = 0.62 \[ -\frac{2 \sin ^2(x) \left (3 \cot (x)+5 \sin (x) \cos (x)+5 \sin ^{\frac{5}{2}}(x) F\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )\right )}{21 \left (a \sin ^3(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sin[x]^2*(3*Cot[x] + 5*Cos[x]*Sin[x] + 5*EllipticF[(Pi - 2*x)/4, 2]*Sin[x]^(5/2)))/(21*(a*Sin[x]^3)^(3/2))

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Maple [C]  time = 0.283, size = 360, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(cos(x)+1)^2*(-1+cos(x))^2*(5*I*2^(1/2)*sin(x)*cos(x)^3*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+s
in(x)+I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))
+5*I*2^(1/2)*sin(x)*cos(x)^2*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*(-I*(-1+co
s(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-5*I*2^(1/2)*sin(x)*cos(x)*((I*co
s(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*c
os(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-5*I*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*2^(1/2)*((-I*cos(x)+sin(x)+I
)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*sin(x)
-10*cos(x)^3+16*cos(x))/(a*sin(x)^3)^(3/2)/sin(x)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(x)^3)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{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}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-(a*cos(x)^2 - a)*sin(x))/(a^2*cos(x)^6 - 3*a^2*cos(x)^4 + 3*a^2*cos(x)^2 - a^2), x)

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

[Out]

Integral((a*sin(x)**3)**(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(x)^3)^(-3/2), x)