\(\int \frac {1}{(a \csc ^2(x))^{5/2}} \, dx\) [53]

Optimal result

Integrand size = 10, antiderivative size = 55 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \sqrt {a \csc ^2(x)} \sin (x)}{240 a^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {\sin \left (x \right )^{3} \left (8+3 \cos \left (x \right )^{3}-6 \cos \left (x \right )^{2}-\cos \left (x \right )\right ) \sqrt {4}}{30 \left (\cos \left (x \right )-1\right )^{2} \sqrt {a \csc \left (x \right )^{2}}\, a^{2}}\)	\(45\)
risch	\(-\frac {i {\mathrm e}^{6 i x}}{160 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i {\mathrm e}^{2 i x}}{16 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i}{16 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}+\frac {5 i {\mathrm e}^{-2 i x}}{96 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {11 i \cos \left (4 x \right )}{240 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}-\frac {7 \sin \left (4 x \right )}{120 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}\)	\(228\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {{\left (3 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{15 \, a^{3}} \]

[In]

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

[Out]

-1/15*(3*cos(x)^5 - 10*cos(x)^3 + 15*cos(x))*sqrt(-a/(cos(x)^2 - 1))*sin(x)/a^3

Sympy [A] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=- \frac {8 \cot ^{5}{\left (x \right )}}{15 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {4 \cot ^{3}{\left (x \right )}}{3 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((a*csc(x)^2)^(-5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=\frac {16 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{15 \, a^{\frac {5}{2}}} - \frac {16 \, {\left (\frac {5 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {10 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}}{15 \, a^{\frac {5}{2}} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{5} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a/sin(x)^2)^(5/2), x)