3.12 \(\int \frac{1}{(a \sin ^3(x))^{5/2}} \, dx\)
Optimal. Leaf size=123 \[ -\frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}+\frac{154 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}} \]
[Out]
(-154*Cot[x])/(585*a^2*Sqrt[a*Sin[x]^3]) - (22*Cot[x]*Csc[x]^2)/(117*a^2*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x]^
4)/(13*a^2*Sqrt[a*Sin[x]^3]) - (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Sin[x]^3]) + (154*EllipticE[Pi/4 - x/2, 2]*
Sin[x]^(3/2))/(195*a^2*Sqrt[a*Sin[x]^3])
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Rubi [A] time = 0.0420212, antiderivative size = 123, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used =
{3207, 2636, 2639} \[ -\frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}+\frac{154 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}} \]
Antiderivative was successfully verified.
[In]
Int[(a*Sin[x]^3)^(-5/2),x]
[Out]
(-154*Cot[x])/(585*a^2*Sqrt[a*Sin[x]^3]) - (22*Cot[x]*Csc[x]^2)/(117*a^2*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x]^
4)/(13*a^2*Sqrt[a*Sin[x]^3]) - (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Sin[x]^3]) + (154*EllipticE[Pi/4 - x/2, 2]*
Sin[x]^(3/2))/(195*a^2*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 2639
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(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 )^{5/2}} \, dx &=\frac{\sin ^{\frac{3}{2}}(x) \int \frac{1}{\sin ^{\frac{15}{2}}(x)} \, dx}{a^2 \sqrt{a \sin ^3(x)}}\\ &=-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}+\frac{\left (11 \sin ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sin ^{\frac{11}{2}}(x)} \, dx}{13 a^2 \sqrt{a \sin ^3(x)}}\\ &=-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}+\frac{\left (77 \sin ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sin ^{\frac{7}{2}}(x)} \, dx}{117 a^2 \sqrt{a \sin ^3(x)}}\\ &=-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}+\frac{\left (77 \sin ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sin ^{\frac{3}{2}}(x)} \, dx}{195 a^2 \sqrt{a \sin ^3(x)}}\\ &=-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}-\frac{154 \cos (x) \sin (x)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{\left (77 \sin ^{\frac{3}{2}}(x)\right ) \int \sqrt{\sin (x)} \, dx}{195 a^2 \sqrt{a \sin ^3(x)}}\\ &=-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}-\frac{154 \cos (x) \sin (x)}{195 a^2 \sqrt{a \sin ^3(x)}}+\frac{154 E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sin ^{\frac{3}{2}}(x)}{195 a^2 \sqrt{a \sin ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.208115, size = 60, normalized size = 0.49 \[ -\frac{2 \left (231 \sin (x) \cos (x)+\cot (x) \left (45 \csc ^4(x)+55 \csc ^2(x)+77\right )-231 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )\right )}{585 a^2 \sqrt{a \sin ^3(x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*Sin[x]^3)^(-5/2),x]
[Out]
(-2*(Cot[x]*(77 + 55*Csc[x]^2 + 45*Csc[x]^4) + 231*Cos[x]*Sin[x] - 231*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)
))/(585*a^2*Sqrt[a*Sin[x]^3])
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Maple [C] time = 0.248, size = 1301, normalized size = 10.6 \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)^(5/2),x)
[Out]
-1/585*(462*2^(1/2)*cos(x)^7*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(
x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-231*2^(1/2)*cos(x)^7*(-I*(-1+cos
(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)
+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))+462*2^(1/2)*cos(x)^6*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/
sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-23
1*2^(1/2)*cos(x)^6*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(
x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-1386*2^(1/2)*cos(x)^5*(-I*(-1+cos(x))/sin(
x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I
)/sin(x))^(1/2),1/2*2^(1/2))+693*2^(1/2)*cos(x)^5*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(
1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-1386*2^(1/2
)*cos(x)^4*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2
)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))+693*2^(1/2)*cos(x)^4*(-I*(-1+cos(x))/sin(x))^(1/2)
*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))
^(1/2),1/2*2^(1/2))+1386*2^(1/2)*cos(x)^3*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-
I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-693*2^(1/2)*cos(x)^
3*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*Ellipti
cF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))+1386*2^(1/2)*cos(x)^2*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos
(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1
/2*2^(1/2))-693*2^(1/2)*cos(x)^2*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+
sin(x)+I)/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-462*2^(1/2)*cos(x)*((-I*cos(
x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*((I*cos(x)+sin(x)-I)/sin(
x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)+231*2^(1/2)*cos(x)*((-I*cos(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*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-46
2*cos(x)^6-462*2^(1/2)*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^
(1/2))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)+231*2^(1/2)*((-I*cos(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*cos(x)+sin(x)-I)/sin(
x))^(1/2),1/2*2^(1/2))+154*cos(x)^5+1386*cos(x)^4-418*cos(x)^3-1386*cos(x)^2+354*cos(x)+462)*sin(x)/(a*sin(x)^
3)^(5/2)
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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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="maxima")
[Out]
integrate((a*sin(x)^3)^(-5/2), 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 )}}{{\left (a^{3} \cos \left (x\right )^{8} - 4 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} - 4 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )} \sin \left (x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="fricas")
[Out]
integral(sqrt(-(a*cos(x)^2 - a)*sin(x))/((a^3*cos(x)^8 - 4*a^3*cos(x)^6 + 6*a^3*cos(x)^4 - 4*a^3*cos(x)^2 + a^
3)*sin(x)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*sin(x)**3)**(5/2),x)
[Out]
Timed out
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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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="giac")
[Out]
integrate((a*sin(x)^3)^(-5/2), x)