\(\int \frac {1}{(a \csc ^4(x))^{5/2}} \, dx\) [67]

Optimal result

Integrand size = 10, antiderivative size = 132 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}+\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}} \]

[Out]

-63/256*cot(x)/a^2/(a*csc(x)^4)^(1/2)+63/256*x*csc(x)^2/a^2/(a*csc(x)^4)^(1/2)-21/128*cos(x)*sin(x)/a^2/(a*csc
(x)^4)^(1/2)-21/160*cos(x)*sin(x)^3/a^2/(a*csc(x)^4)^(1/2)-9/80*cos(x)*sin(x)^5/a^2/(a*csc(x)^4)^(1/2)-1/10*co
s(x)*sin(x)^7/a^2/(a*csc(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8} \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {\sin ^7(x) \cos (x)}{10 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \sin ^5(x) \cos (x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin ^3(x) \cos (x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin (x) \cos (x)}{128 a^2 \sqrt {a \csc ^4(x)}} \]

[In]

Int[(a*Csc[x]^4)^(-5/2),x]

[Out]

(-63*Cot[x])/(256*a^2*Sqrt[a*Csc[x]^4]) + (63*x*Csc[x]^2)/(256*a^2*Sqrt[a*Csc[x]^4]) - (21*Cos[x]*Sin[x])/(128
*a^2*Sqrt[a*Csc[x]^4]) - (21*Cos[x]*Sin[x]^3)/(160*a^2*Sqrt[a*Csc[x]^4]) - (9*Cos[x]*Sin[x]^5)/(80*a^2*Sqrt[a*
Csc[x]^4]) - (Cos[x]*Sin[x]^7)/(10*a^2*Sqrt[a*Csc[x]^4])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^2(x) \int \sin ^{10}(x) \, dx}{a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (9 \csc ^2(x)\right ) \int \sin ^8(x) \, dx}{10 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int \sin ^6(x) \, dx}{80 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (21 \csc ^2(x)\right ) \int \sin ^4(x) \, dx}{32 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int \sin ^2(x) \, dx}{128 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}+\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \csc ^4(x)} \sin ^2(x) (2520 x-2100 \sin (2 x)+600 \sin (4 x)-150 \sin (6 x)+25 \sin (8 x)-2 \sin (10 x))}{10240 a^3} \]

[In]

Integrate[(a*Csc[x]^4)^(-5/2),x]

[Out]

(Sqrt[a*Csc[x]^4]*Sin[x]^2*(2520*x - 2100*Sin[2*x] + 600*Sin[4*x] - 150*Sin[6*x] + 25*Sin[8*x] - 2*Sin[10*x]))
/(10240*a^3)

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.46

method	result	size
default	\(-\frac {\left (128 \cos \left (x \right )^{8} \cot \left (x \right )-656 \cos \left (x \right )^{6} \cot \left (x \right )+1368 \cos \left (x \right )^{4} \cot \left (x \right )-1490 \cos \left (x \right )^{2} \cot \left (x \right )+965 \cot \left (x \right )-315 \csc \left (x \right )^{2} x \right ) \sqrt {16}}{5120 \sqrt {a \csc \left (x \right )^{4}}\, a^{2}}\)	\(61\)
risch	\(-\frac {63 \,{\mathrm e}^{2 i x} x}{256 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {i {\mathrm e}^{12 i x}}{10240 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {5 i {\mathrm e}^{10 i x}}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {105 i {\mathrm e}^{4 i x}}{1024 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {105 i}{1024 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {15 i {\mathrm e}^{-2 i x}}{512 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {15 i {\mathrm e}^{-4 i x}}{2048 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {37 i \cos \left (8 x \right )}{5120 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {19 \sin \left (8 x \right )}{2560 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {115 i \cos \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {125 \sin \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}\)	\(409\)

[In]

int(1/(a*csc(x)^4)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/5120/(a*csc(x)^4)^(1/2)/a^2*(128*cos(x)^8*cot(x)-656*cos(x)^6*cot(x)+1368*cos(x)^4*cot(x)-1490*cos(x)^2*cot
(x)+965*cot(x)-315*csc(x)^2*x)*16^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {{\left (315 \, x \cos \left (x\right )^{2} - {\left (128 \, \cos \left (x\right )^{11} - 784 \, \cos \left (x\right )^{9} + 2024 \, \cos \left (x\right )^{7} - 2858 \, \cos \left (x\right )^{5} + 2455 \, \cos \left (x\right )^{3} - 965 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 315 \, x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{1280 \, a^{3}} \]

[In]

integrate(1/(a*csc(x)^4)^(5/2),x, algorithm="fricas")

[Out]

-1/1280*(315*x*cos(x)^2 - (128*cos(x)^11 - 784*cos(x)^9 + 2024*cos(x)^7 - 2858*cos(x)^5 + 2455*cos(x)^3 - 965*
cos(x))*sin(x) - 315*x)*sqrt(a/(cos(x)^4 - 2*cos(x)^2 + 1))/a^3

Sympy [F]

\[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \csc ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(a*csc(x)**4)**(5/2),x)

[Out]

Integral((a*csc(x)**4)**(-5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {965 \, \tan \left (x\right )^{9} + 2370 \, \tan \left (x\right )^{7} + 2688 \, \tan \left (x\right )^{5} + 1470 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{1280 \, {\left (a^{\frac {5}{2}} \tan \left (x\right )^{10} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{8} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{6} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{4} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{2} + a^{\frac {5}{2}}\right )}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \]

[In]

integrate(1/(a*csc(x)^4)^(5/2),x, algorithm="maxima")

[Out]

-1/1280*(965*tan(x)^9 + 2370*tan(x)^7 + 2688*tan(x)^5 + 1470*tan(x)^3 + 315*tan(x))/(a^(5/2)*tan(x)^10 + 5*a^(
5/2)*tan(x)^8 + 10*a^(5/2)*tan(x)^6 + 10*a^(5/2)*tan(x)^4 + 5*a^(5/2)*tan(x)^2 + a^(5/2)) + 63/256*x/a^(5/2)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a*csc(x)^4)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^4}\right )}^{5/2}} \,d x \]

[In]

int(1/(a/sin(x)^4)^(5/2),x)

[Out]

int(1/(a/sin(x)^4)^(5/2), x)