3.6 \(\int \frac{1}{(a \sin ^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}} \]

[Out]

-Cot[x]/(4*a*(a*Sin[x]^2)^(3/2)) - (3*Cot[x])/(8*a^2*Sqrt[a*Sin[x]^2]) - (3*ArcTanh[Cos[x]]*Sin[x])/(8*a^2*Sqr
t[a*Sin[x]^2])

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Rubi [A]  time = 0.0302122, antiderivative size = 61, 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 = {3204, 3207, 3770} \[ -\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sin[x]^2)^(-5/2),x]

[Out]

-Cot[x]/(4*a*(a*Sin[x]^2)^(3/2)) - (3*Cot[x])/(8*a^2*Sqrt[a*Sin[x]^2]) - (3*ArcTanh[Cos[x]]*Sin[x])/(8*a^2*Sqr
t[a*Sin[x]^2])

Rule 3204

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1))/(b*f*(
2*p + 1)), x] + Dist[(2*(p + 1))/(b*(2*p + 1)), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[p, -1]

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \sin ^2(x)\right )^{5/2}} \, dx &=-\frac{\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}+\frac{3 \int \frac{1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx}{4 a}\\ &=-\frac{\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}+\frac{3 \int \frac{1}{\sqrt{a \sin ^2(x)}} \, dx}{8 a^2}\\ &=-\frac{\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}+\frac{(3 \sin (x)) \int \csc (x) \, dx}{8 a^2 \sqrt{a \sin ^2(x)}}\\ &=-\frac{\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{3 \tanh ^{-1}(\cos (x)) \sin (x)}{8 a^2 \sqrt{a \sin ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.194872, size = 77, normalized size = 1.26 \[ -\frac{\csc (x) \sqrt{a \sin ^2(x)} \left (\csc ^4\left (\frac{x}{2}\right )+6 \csc ^2\left (\frac{x}{2}\right )-\sec ^4\left (\frac{x}{2}\right )-6 \sec ^2\left (\frac{x}{2}\right )+24 \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )\right )}{64 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[x]^2)^(-5/2),x]

[Out]

-(Csc[x]*(6*Csc[x/2]^2 + Csc[x/2]^4 + 24*(Log[Cos[x/2]] - Log[Sin[x/2]]) - 6*Sec[x/2]^2 - Sec[x/2]^4)*Sqrt[a*S
in[x]^2])/(64*a^3)

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Maple [A]  time = 1.156, size = 89, normalized size = 1.5 \begin{align*} -{\frac{1}{8\, \left ( \sin \left ( x \right ) \right ) ^{3}\cos \left ( x \right ) }\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( 3\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}+a}{\sin \left ( x \right ) }} \right ) \left ( \sin \left ( x \right ) \right ) ^{4}a+3\,\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{2}\sqrt{a}+2\,\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a^(7/2)/sin(x)^3*(a*cos(x)^2)^(1/2)*(3*ln(2*(a^(1/2)*(a*cos(x)^2)^(1/2)+a)/sin(x))*sin(x)^4*a+3*(a*cos(x)
^2)^(1/2)*sin(x)^2*a^(1/2)+2*a^(1/2)*(a*cos(x)^2)^(1/2))/cos(x)/(a*sin(x)^2)^(1/2)

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Maxima [B]  time = 2.55918, size = 1257, normalized size = 20.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/8*(3*(2*(4*cos(6*x) - 6*cos(4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*
cos(6*x) - 16*cos(6*x)^2 + 12*(4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*si
n(4*x) + 2*sin(2*x))*sin(8*x) - sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*
x)^2 + 48*sin(4*x)*sin(2*x) - 16*sin(2*x)^2 + 8*cos(2*x) - 1)*arctan2(sin(x), cos(x) + 1) - 3*(2*(4*cos(6*x) -
6*cos(4*x) + 4*cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 8*(6*cos(4*x) - 4*cos(2*x) + 1)*cos(6*x) - 16*cos(6*x)^2
+ 12*(4*cos(2*x) - 1)*cos(4*x) - 36*cos(4*x)^2 - 16*cos(2*x)^2 + 4*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*sin
(8*x) - sin(8*x)^2 + 16*(3*sin(4*x) - 2*sin(2*x))*sin(6*x) - 16*sin(6*x)^2 - 36*sin(4*x)^2 + 48*sin(4*x)*sin(2
*x) - 16*sin(2*x)^2 + 8*cos(2*x) - 1)*arctan2(sin(x), cos(x) - 1) + 2*(3*sin(7*x) - 11*sin(5*x) - 11*sin(3*x)
+ 3*sin(x))*cos(8*x) + 12*(2*sin(6*x) - 3*sin(4*x) + 2*sin(2*x))*cos(7*x) + 8*(11*sin(5*x) + 11*sin(3*x) - 3*s
in(x))*cos(6*x) + 44*(3*sin(4*x) - 2*sin(2*x))*cos(5*x) - 12*(11*sin(3*x) - 3*sin(x))*cos(4*x) - 2*(3*cos(7*x)
- 11*cos(5*x) - 11*cos(3*x) + 3*cos(x))*sin(8*x) - 6*(4*cos(6*x) - 6*cos(4*x) + 4*cos(2*x) - 1)*sin(7*x) - 8*
(11*cos(5*x) + 11*cos(3*x) - 3*cos(x))*sin(6*x) - 22*(6*cos(4*x) - 4*cos(2*x) + 1)*sin(5*x) + 12*(11*cos(3*x)
- 3*cos(x))*sin(4*x) + 22*(4*cos(2*x) - 1)*sin(3*x) - 88*cos(3*x)*sin(2*x) + 24*cos(x)*sin(2*x) - 24*cos(2*x)*
sin(x) + 6*sin(x))*sqrt(-a)/(a^3*cos(8*x)^2 + 16*a^3*cos(6*x)^2 + 36*a^3*cos(4*x)^2 + 16*a^3*cos(2*x)^2 + a^3*
sin(8*x)^2 + 16*a^3*sin(6*x)^2 + 36*a^3*sin(4*x)^2 - 48*a^3*sin(4*x)*sin(2*x) + 16*a^3*sin(2*x)^2 - 8*a^3*cos(
2*x) + a^3 - 2*(4*a^3*cos(6*x) - 6*a^3*cos(4*x) + 4*a^3*cos(2*x) - a^3)*cos(8*x) - 8*(6*a^3*cos(4*x) - 4*a^3*c
os(2*x) + a^3)*cos(6*x) - 12*(4*a^3*cos(2*x) - a^3)*cos(4*x) - 4*(2*a^3*sin(6*x) - 3*a^3*sin(4*x) + 2*a^3*sin(
2*x))*sin(8*x) - 16*(3*a^3*sin(4*x) - 2*a^3*sin(2*x))*sin(6*x))

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Fricas [A]  time = 1.70054, size = 221, normalized size = 3.62 \begin{align*} \frac{\sqrt{-a \cos \left (x\right )^{2} + a}{\left (6 \, \cos \left (x\right )^{3} + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) - 10 \, \cos \left (x\right )\right )}}{16 \,{\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*sqrt(-a*cos(x)^2 + a)*(6*cos(x)^3 + 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-(cos(x) - 1)/(cos(x) + 1)) - 10*co
s(x))/((a^3*cos(x)^4 - 2*a^3*cos(x)^2 + a^3)*sin(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)**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.29953, size = 108, normalized size = 1.77 \begin{align*} \frac{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{4} + 8 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{2} + \frac{12 \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2}\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )} - \frac{18 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 8 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{4}}}{64 \, a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sin(x)^2)^(5/2),x, algorithm="giac")

[Out]

1/64*(sgn(tan(1/2*x))*tan(1/2*x)^4 + 8*sgn(tan(1/2*x))*tan(1/2*x)^2 + 12*log(tan(1/2*x)^2)/sgn(tan(1/2*x)) - (
18*tan(1/2*x)^4 + 8*tan(1/2*x)^2 + 1)/(sgn(tan(1/2*x))*tan(1/2*x)^4))/a^(5/2)