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

Optimal. Leaf size=42 \[ -\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 a \sqrt{a \sin ^2(x)}} \]

[Out]

-Cot[x]/(2*a*Sqrt[a*Sin[x]^2]) - (ArcTanh[Cos[x]]*Sin[x])/(2*a*Sqrt[a*Sin[x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0242362, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3204, 3207, 3770} \[ -\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 a \sqrt{a \sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cot[x]/(2*a*Sqrt[a*Sin[x]^2]) - (ArcTanh[Cos[x]]*Sin[x])/(2*a*Sqrt[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 )^{3/2}} \, dx &=-\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}+\frac{\int \frac{1}{\sqrt{a \sin ^2(x)}} \, dx}{2 a}\\ &=-\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}+\frac{\sin (x) \int \csc (x) \, dx}{2 a \sqrt{a \sin ^2(x)}}\\ &=-\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}-\frac{\tanh ^{-1}(\cos (x)) \sin (x)}{2 a \sqrt{a \sin ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.058478, size = 55, normalized size = 1.31 \[ -\frac{\sin ^3(x) \left (\csc ^2\left (\frac{x}{2}\right )-\sec ^2\left (\frac{x}{2}\right )-4 \log \left (\sin \left (\frac{x}{2}\right )\right )+4 \log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{8 \left (a \sin ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 1.116, size = 70, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,\sin \left ( x \right ) \cos \left ( x \right ) }\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \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 ) ^{2}a+\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \right ){a}^{-{\frac{5}{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)^(3/2),x)

[Out]

-1/2/a^(5/2)/sin(x)*(a*cos(x)^2)^(1/2)*(ln(2*(a^(1/2)*(a*cos(x)^2)^(1/2)+a)/sin(x))*sin(x)^2*a+a^(1/2)*(a*cos(
x)^2)^(1/2))/cos(x)/(a*sin(x)^2)^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 1.64417, size = 424, normalized size = 10.1 \begin{align*} -\frac{{\left ({\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )\right )} \sqrt{-a}}{2 \,{\left (a^{2} \cos \left (4 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right )^{2} + a^{2} \sin \left (4 \, x\right )^{2} - 4 \, a^{2} \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a^{2} \sin \left (2 \, x\right )^{2} - 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2} - 2 \,{\left (2 \, a^{2} \cos \left (2 \, x\right ) - a^{2}\right )} \cos \left (4 \, x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)
^2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 -
sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) - 1) + 2*(sin(3*x) +
sin(x))*cos(4*x) - 2*(cos(3*x) + cos(x))*sin(4*x) - 2*(2*cos(2*x) - 1)*sin(3*x) + 4*cos(3*x)*sin(2*x) + 4*cos(
x)*sin(2*x) - 4*cos(2*x)*sin(x) + 2*sin(x))*sqrt(-a)/(a^2*cos(4*x)^2 + 4*a^2*cos(2*x)^2 + a^2*sin(4*x)^2 - 4*a
^2*sin(4*x)*sin(2*x) + 4*a^2*sin(2*x)^2 - 4*a^2*cos(2*x) + a^2 - 2*(2*a^2*cos(2*x) - a^2)*cos(4*x))

________________________________________________________________________________________

Fricas [A]  time = 1.6649, size = 158, normalized size = 3.76 \begin{align*} \frac{\sqrt{-a \cos \left (x\right )^{2} + a}{\left ({\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) + 2 \, \cos \left (x\right )\right )}}{4 \,{\left (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)^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*sqrt(-a*cos(x)^2 + a)*((cos(x)^2 - 1)*log(-(cos(x) - 1)/(cos(x) + 1)) + 2*cos(x))/((a^2*cos(x)^2 - a^2)*si
n(x))

________________________________________________________________________________________

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

[Out]

Integral((a*sin(x)**2)**(-3/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.27683, size = 82, normalized size = 1.95 \begin{align*} \frac{\frac{\tan \left (\frac{1}{2} \, x\right )^{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )} + \frac{2 \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2}\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )} - \frac{2 \, \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 )^{2}}}{8 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(tan(1/2*x)^2/sgn(tan(1/2*x)) + 2*log(tan(1/2*x)^2)/sgn(tan(1/2*x)) - (2*tan(1/2*x)^2 + 1)/(sgn(tan(1/2*x)
)*tan(1/2*x)^2))/a^(3/2)