\(\int \frac {1}{(a \csc ^2(x))^{3/2}} \, dx\) [52]

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}} \]

[Out]

-1/3*cot(x)/(a*csc(x)^2)^(3/2)-2/3*cot(x)/a/(a*csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}}-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}} \]

[In]

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

[Out]

-1/3*Cot[x]/(a*Csc[x]^2)^(3/2) - (2*Cot[x])/(3*a*Sqrt[a*Csc[x]^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\frac {(-9 \cos (x)+\cos (3 x)) \csc ^3(x)}{12 \left (a \csc ^2(x)\right )^{3/2}} \]

[In]

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

[Out]

((-9*Cos[x] + Cos[3*x])*Csc[x]^3)/(12*(a*Csc[x]^2)^(3/2))

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {\sin \left (x \right ) \left (-2+\cos \left (x \right )^{2}-\cos \left (x \right )\right ) \sqrt {4}}{6 \left (\cos \left (x \right )-1\right ) \sqrt {a \csc \left (x \right )^{2}}\, a}\)	\(35\)
risch	\(\frac {i {\mathrm e}^{4 i x}}{24 a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {3 i {\mathrm e}^{2 i x}}{8 a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {3 i}{8 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a}+\frac {i {\mathrm e}^{-2 i x}}{24 a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(153\)

[In]

int(1/(a*csc(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*sin(x)*(-2+cos(x)^2-cos(x))/(cos(x)-1)/(a*csc(x)^2)^(1/2)/a*4^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\frac {{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{3 \, a^{2}} \]

[In]

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

[Out]

1/3*(cos(x)^3 - 3*cos(x))*sqrt(-a/(cos(x)^2 - 1))*sin(x)/a^2

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=- \frac {2 \cot ^{3}{\left (x \right )}}{3 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} \]

[In]

integrate(1/(a*csc(x)**2)**(3/2),x)

[Out]

-2*cot(x)**3/(3*(a*csc(x)**2)**(3/2)) - cot(x)/(a*csc(x)**2)**(3/2)

Maxima [F]

\[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{2}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csc(x)^2)^(-3/2), x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\frac {4 \, {\left (\frac {\mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} - \frac {\frac {3 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1}{\sqrt {a} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )}\right )}}{3 \, a} \]

[In]

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

[Out]

4/3*(sgn(sin(x))/sqrt(a) - (3*(cos(x) - 1)/(cos(x) + 1) - 1)/(sqrt(a)*((cos(x) - 1)/(cos(x) + 1) - 1)^3*sgn(si
n(x))))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

int(1/(a/sin(x)^2)^(3/2), x)