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]

-154/585*cot(x)/a^2/(a*sin(x)^3)^(1/2)-22/117*cot(x)*csc(x)^2/a^2/(a*sin(x)^3)^(1/2)-2/13*cot(x)*csc(x)^4/a^2/
(a*sin(x)^3)^(1/2)-154/195*cos(x)*sin(x)/a^2/(a*sin(x)^3)^(1/2)+154/195*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/4*Pi
+1/2*x)*EllipticE(cos(1/4*Pi+1/2*x),2^(1/2))*sin(x)^(3/2)/a^2/(a*sin(x)^3)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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
]])

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*}

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Mathematica [A]  time = 0.22, 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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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}}{{\left (a^{3} \cos \relax (x)^{8} - 4 \, a^{3} \cos \relax (x)^{6} + 6 \, a^{3} \cos \relax (x)^{4} - 4 \, a^{3} \cos \relax (x)^{2} + a^{3}\right )} \sin \relax (x)}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sin \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [C]  time = 0.40, size = 1301, normalized size = 10.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sin(x)^3)^(5/2),x)

[Out]

1/585*(231*(-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))*2^(1/2)*cos(x)^7-462*(-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))*2^(1/2)*cos(x)^7+231*(-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))*2^(1/2)*cos(x)^6-462
*(-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)*Elliptic
E(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*2^(1/2)*cos(x)^6-693*(-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))*2^(1/2)*cos(x)^5+1386*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*((-I*cos(x)+s
in(x)+I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*2^(1/2)*cos(x)^5-693*(-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))*2^(1/2)*cos(x)^4+1386*(-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)
)*2^(1/2)*cos(x)^4+693*(-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))*2^(1/2)*cos(x)^3-1386*(-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))*2^(1/2)*cos(x)^3+693*(-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))*2^(1/2)
*cos(x)^2-1386*(-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))*2^(1/2)*cos(x)^2-231*((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))*2^(1/2)*cos(x)+462*((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)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*2^(1/2)*cos(x)+462
*cos(x)^6-231*((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))*2^(1/2)+462*((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)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2)
,1/2*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sin \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,{\sin \relax (x)}^3\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sin(x)^3)^(5/2),x)

[Out]

int(1/(a*sin(x)^3)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sin ^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)**3)**(5/2),x)

[Out]

Integral((a*sin(x)**3)**(-5/2), x)

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