\(\int \frac {1}{\csc ^2(x)^{5/2}} \, dx\) [45]

Optimal result

Integrand size = 8, antiderivative size = 43 \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {8 \cot (x)}{15 \sqrt {\csc ^2(x)}} \]

[Out]

-1/5*cot(x)/(csc(x)^2)^(5/2)-4/15*cot(x)/(csc(x)^2)^(3/2)-8/15*cot(x)/(csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 198, 197} \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=-\frac {8 \cot (x)}{15 \sqrt {\csc ^2(x)}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}} \]

[In]

Int[(Csc[x]^2)^(-5/2),x]

[Out]

-1/5*Cot[x]/(Csc[x]^2)^(5/2) - (4*Cot[x])/(15*(Csc[x]^2)^(3/2)) - (8*Cot[x])/(15*Sqrt[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}& = -\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4}{5} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {8}{15} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {8 \cot (x)}{15 \sqrt {\csc ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=-\frac {(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x)}{240 \sqrt {\csc ^2(x)}} \]

[In]

Integrate[(Csc[x]^2)^(-5/2),x]

[Out]

-1/240*((150*Cos[x] - 25*Cos[3*x] + 3*Cos[5*x])*Csc[x])/Sqrt[Csc[x]^2]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.49 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {\sin \left (x \right )^{4} \operatorname {csgn}\left (\csc \left (x \right )\right ) \left (8+3 \cos \left (x \right )^{3}-6 \cos \left (x \right )^{2}-\cos \left (x \right )\right ) \sqrt {4}}{30 \left (\cos \left (x \right )-1\right )^{2}}\)	\(37\)
risch	\(-\frac {i {\mathrm e}^{6 i x}}{160 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i {\mathrm e}^{2 i x}}{16 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {5 i}{16 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {5 i {\mathrm e}^{-2 i x}}{96 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {11 i \cos \left (4 x \right )}{240 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {7 \sin \left (4 x \right )}{120 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(204\)

[In]

int(1/(csc(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/30*sin(x)^4*csgn(csc(x))*(8+3*cos(x)^3-6*cos(x)^2-cos(x))/(cos(x)-1)^2*4^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=-\frac {1}{5} \, \cos \left (x\right )^{5} + \frac {2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

[In]

integrate(1/(csc(x)^2)^(5/2),x, algorithm="fricas")

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

Sympy [A] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=- \frac {8 \cot ^{5}{\left (x \right )}}{15 \left (\csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {4 \cot ^{3}{\left (x \right )}}{3 \left (\csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {\cot {\left (x \right )}}{\left (\csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} \]

[In]

integrate(1/(csc(x)**2)**(5/2),x)

[Out]

-8*cot(x)**5/(15*(csc(x)**2)**(5/2)) - 4*cot(x)**3/(3*(csc(x)**2)**(5/2)) - cot(x)/(csc(x)**2)**(5/2)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=-\frac {1}{80} \, \cos \left (5 \, x\right ) + \frac {5}{48} \, \cos \left (3 \, x\right ) - \frac {5}{8} \, \cos \left (x\right ) \]

[In]

integrate(1/(csc(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/80*cos(5*x) + 5/48*cos(3*x) - 5/8*cos(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=-\frac {16 \, {\left (\frac {5 \, {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \frac {10 \, {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \mathrm {sgn}\left (\sin \left (x\right )\right )\right )}}{15 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{5}} + \frac {16}{15} \, \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[In]

integrate(1/(csc(x)^2)^(5/2),x, algorithm="giac")

[Out]

-16/15*(5*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - 10*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 - sgn(sin(x)))/
((cos(x) - 1)/(cos(x) + 1) - 1)^5 + 16/15*sgn(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\csc ^2(x)^{5/2}} \, dx=\int \frac {1}{{\left (\frac {1}{{\sin \left (x\right )}^2}\right )}^{5/2}} \,d x \]

[In]

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

[Out]

int(1/(1/sin(x)^2)^(5/2), x)