\(\int \frac {1}{(a \csc ^2(x))^{7/2}} \, dx\) [54]

Optimal result

Integrand size = 10, antiderivative size = 74 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=-\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {16 \cot (x)}{35 a^3 \sqrt {a \csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=-\frac {16 \cot (x)}{35 a^3 \sqrt {a \csc ^2(x)}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \sqrt {a \csc ^2(x)} \sin (x)}{2240 a^4} \]

[In]

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

[Out]

((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Sqrt[a*Csc[x]^2]*Sin[x])/(2240*a^4)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {\sin \left (x \right )^{5} \left (-16+5 \cos \left (x \right )^{4}-15 \cos \left (x \right )^{3}+9 \cos \left (x \right )^{2}+13 \cos \left (x \right )\right ) \sqrt {4}}{70 \left (\cos \left (x \right )-1\right )^{3} \sqrt {a \csc \left (x \right )^{2}}\, a^{3}}\)	\(51\)
risch	\(\frac {i {\mathrm e}^{8 i x}}{896 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i {\mathrm e}^{2 i x}}{128 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i}{128 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {7 i {\mathrm e}^{-2 i x}}{128 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {11 i \cos \left (6 x \right )}{1120 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {27 \sin \left (6 x \right )}{2240 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {7 i \cos \left (4 x \right )}{160 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}-\frac {21 \sin \left (4 x \right )}{320 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}\)	\(303\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {{\left (5 \, \cos \left (x\right )^{7} - 21 \, \cos \left (x\right )^{5} + 35 \, \cos \left (x\right )^{3} - 35 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{35 \, a^{4}} \]

[In]

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

[Out]

1/35*(5*cos(x)^7 - 21*cos(x)^5 + 35*cos(x)^3 - 35*cos(x))*sqrt(-a/(cos(x)^2 - 1))*sin(x)/a^4

Sympy [A] (verification not implemented)

Time = 10.63 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=- \frac {16 \cot ^{7}{\left (x \right )}}{35 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\left (x \right )}}{5 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((a*csc(x)^2)^(-7/2), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {32 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{35 \, a^{\frac {7}{2}}} - \frac {32 \, {\left (\frac {7 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {21 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )}}{35 \, a^{\frac {7}{2}} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{7} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a/sin(x)^2)^(7/2), x)