Integrand size = 10, antiderivative size = 74 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=-\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {16 \cot (x)}{35 a^3 \sqrt {a \csc ^2(x)}} \] Output:
-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)
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \sqrt {a \csc ^2(x)} \sin (x)}{2240 a^4} \] Input:
Integrate[(a*Csc[x]^2)^(-7/2),x]
Output:
((-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)
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4610, 209, 209, 209, 208}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sec \left (x+\frac {\pi }{2}\right )^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle -a \int \frac {1}{\left (a \cot ^2(x)+a\right )^{9/2}}d\cot (x)\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -a \left (\frac {6 \int \frac {1}{\left (a \cot ^2(x)+a\right )^{7/2}}d\cot (x)}{7 a}+\frac {\cot (x)}{7 a \left (a \cot ^2(x)+a\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -a \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (a \cot ^2(x)+a\right )^{5/2}}d\cot (x)}{5 a}+\frac {\cot (x)}{5 a \left (a \cot ^2(x)+a\right )^{5/2}}\right )}{7 a}+\frac {\cot (x)}{7 a \left (a \cot ^2(x)+a\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -a \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (a \cot ^2(x)+a\right )^{3/2}}d\cot (x)}{3 a}+\frac {\cot (x)}{3 a \left (a \cot ^2(x)+a\right )^{3/2}}\right )}{5 a}+\frac {\cot (x)}{5 a \left (a \cot ^2(x)+a\right )^{5/2}}\right )}{7 a}+\frac {\cot (x)}{7 a \left (a \cot ^2(x)+a\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 208 |
\(\displaystyle -a \left (\frac {6 \left (\frac {4 \left (\frac {2 \cot (x)}{3 a^2 \sqrt {a \cot ^2(x)+a}}+\frac {\cot (x)}{3 a \left (a \cot ^2(x)+a\right )^{3/2}}\right )}{5 a}+\frac {\cot (x)}{5 a \left (a \cot ^2(x)+a\right )^{5/2}}\right )}{7 a}+\frac {\cot (x)}{7 a \left (a \cot ^2(x)+a\right )^{7/2}}\right )\) |
Input:
Int[(a*Csc[x]^2)^(-7/2),x]
Output:
-(a*(Cot[x]/(7*a*(a + a*Cot[x]^2)^(7/2)) + (6*(Cot[x]/(5*a*(a + a*Cot[x]^2 )^(5/2)) + (4*(Cot[x]/(3*a*(a + a*Cot[x]^2)^(3/2)) + (2*Cot[x])/(3*a^2*Sqr t[a + a*Cot[x]^2])))/(5*a)))/(7*a)))
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[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]
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {\sqrt {4}\, \left (5 \cos \left (x \right )^{7}-21 \cos \left (x \right )^{5}+35 \cos \left (x \right )^{3}-35 \cos \left (x \right )-16\right ) \csc \left (x \right )}{70 a^{3} \sqrt {a \csc \left (x \right )^{2}}}\) | \(43\) |
risch | \(\frac {i {\mathrm e}^{8 i x}}{896 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i {\mathrm e}^{2 i x}}{128 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i}{128 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {7 i {\mathrm e}^{-2 i x}}{128 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {11 i \cos \left (6 x \right )}{1120 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {27 \sin \left (6 x \right )}{2240 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {7 i \cos \left (4 x \right )}{160 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}-\frac {21 \sin \left (4 x \right )}{320 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}\) | \(303\) |
Input:
int(1/(a*csc(x)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/70*4^(1/2)*(5*cos(x)^7-21*cos(x)^5+35*cos(x)^3-35*cos(x)-16)/a^3/(a*csc( x)^2)^(1/2)*csc(x)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {{\left (5 \, \cos \left (x\right )^{7} - 21 \, \cos \left (x\right )^{5} + 35 \, \cos \left (x\right )^{3} - 35 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{35 \, a^{4}} \] Input:
integrate(1/(a*csc(x)^2)^(7/2),x, algorithm="fricas")
Output:
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
Time = 12.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=- \frac {16 \cot ^{7}{\left (x \right )}}{35 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\left (x \right )}}{5 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \] Input:
integrate(1/(a*csc(x)**2)**(7/2),x)
Output:
-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)
\[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{2}\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*csc(x)^2)^(7/2),x, algorithm="maxima")
Output:
integrate((a*csc(x)^2)^(-7/2), x)
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {32 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{35 \, a^{\frac {7}{2}}} - \frac {32 \, {\left (\frac {7 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {21 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )}}{35 \, a^{\frac {7}{2}} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{7} \mathrm {sgn}\left (\sin \left (x\right )\right )} \] Input:
integrate(1/(a*csc(x)^2)^(7/2),x, algorithm="giac")
Output:
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)))
Timed out. \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{7/2}} \,d x \] Input:
int(1/(a/sin(x)^2)^(7/2),x)
Output:
int(1/(a/sin(x)^2)^(7/2), x)
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {\sqrt {a}\, \left (-5 \cos \left (x \right ) \sin \left (x \right )^{6}-6 \cos \left (x \right ) \sin \left (x \right )^{4}-8 \cos \left (x \right ) \sin \left (x \right )^{2}-16 \cos \left (x \right )+16\right )}{35 a^{4}} \] Input:
int(1/(a*csc(x)^2)^(7/2),x)
Output:
(sqrt(a)*( - 5*cos(x)*sin(x)**6 - 6*cos(x)*sin(x)**4 - 8*cos(x)*sin(x)**2 - 16*cos(x) + 16))/(35*a**4)