\(\int \frac {1}{\csc ^2(x)^{7/2}} \, dx\) [46]

Optimal result

Integrand size = 8, antiderivative size = 57 \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}} \]

[Out]

-1/7*cot(x)/(csc(x)^2)^(7/2)-6/35*cot(x)/(csc(x)^2)^(5/2)-8/35*cot(x)/(csc(x)^2)^(3/2)-16/35*cot(x)/(csc(x)^2)
^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 198, 197} \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=-\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}} \]

[In]

Int[(Csc[x]^2)^(-7/2),x]

[Out]

-1/7*Cot[x]/(Csc[x]^2)^(7/2) - (6*Cot[x])/(35*(Csc[x]^2)^(5/2)) - (8*Cot[x])/(35*(Csc[x]^2)^(3/2)) - (16*Cot[x
])/(35*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 )^{9/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6}{7} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {24}{35} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16}{35} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=\frac {(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \csc (x)}{2240 \sqrt {\csc ^2(x)}} \]

[In]

Integrate[(Csc[x]^2)^(-7/2),x]

[Out]

((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Csc[x])/(2240*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 = 43, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\sin \left (x \right )^{6} \operatorname {csgn}\left (\csc \left (x \right )\right ) \left (-16+5 \cos \left (x \right )^{4}-15 \cos \left (x \right )^{3}+9 \cos \left (x \right )^{2}+13 \cos \left (x \right )\right ) \sqrt {4}}{70 \left (\cos \left (x \right )-1\right )^{3}}\)	\(43\)
risch	\(\frac {i {\mathrm e}^{8 i x}}{896 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i {\mathrm e}^{2 i x}}{128 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {35 i}{128 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {7 i {\mathrm e}^{-2 i x}}{128 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {11 i \cos \left (6 x \right )}{1120 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {27 \sin \left (6 x \right )}{2240 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {7 i \cos \left (4 x \right )}{160 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {21 \sin \left (4 x \right )}{320 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\)	\(271\)

[In]

int(1/(csc(x)^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/70*sin(x)^6*csgn(csc(x))*(-16+5*cos(x)^4-15*cos(x)^3+9*cos(x)^2+13*cos(x))/(cos(x)-1)^3*4^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=\frac {1}{7} \, \cos \left (x\right )^{7} - \frac {3}{5} \, \cos \left (x\right )^{5} + \cos \left (x\right )^{3} - \cos \left (x\right ) \]

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/7*cos(x)^7 - 3/5*cos(x)^5 + cos(x)^3 - cos(x)

Sympy [A] (verification not implemented)

Time = 10.57 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=- \frac {16 \cot ^{7}{\left (x \right )}}{35 \left (\csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\left (x \right )}}{5 \left (\csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\left (x \right )}}{\left (\csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {\cot {\left (x \right )}}{\left (\csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \]

[In]

integrate(1/(csc(x)**2)**(7/2),x)

[Out]

-16*cot(x)**7/(35*(csc(x)**2)**(7/2)) - 8*cot(x)**5/(5*(csc(x)**2)**(7/2)) - 2*cot(x)**3/(csc(x)**2)**(7/2) -
cot(x)/(csc(x)**2)**(7/2)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=\frac {1}{448} \, \cos \left (7 \, x\right ) - \frac {7}{320} \, \cos \left (5 \, x\right ) + \frac {7}{64} \, \cos \left (3 \, x\right ) - \frac {35}{64} \, \cos \left (x\right ) \]

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="maxima")

[Out]

1/448*cos(7*x) - 7/320*cos(5*x) + 7/64*cos(3*x) - 35/64*cos(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=-\frac {32 \, {\left (\frac {7 \, {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \frac {21 \, {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (x\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \mathrm {sgn}\left (\sin \left (x\right )\right )\right )}}{35 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{7}} + \frac {32}{35} \, \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[In]

integrate(1/(csc(x)^2)^(7/2),x, algorithm="giac")

[Out]

-32/35*(7*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - 21*(cos(x) - 1)^2*sgn(sin(x))/(cos(x) + 1)^2 + 35*(cos(x) -
1)^3*sgn(sin(x))/(cos(x) + 1)^3 - sgn(sin(x)))/((cos(x) - 1)/(cos(x) + 1) - 1)^7 + 32/35*sgn(sin(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\csc ^2(x)^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{{\sin \left (x\right )}^2}\right )}^{7/2}} \,d x \]

[In]

int(1/(1/sin(x)^2)^(7/2),x)

[Out]

int(1/(1/sin(x)^2)^(7/2), x)