Integrand size = 10, antiderivative size = 60 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {\sqrt {1-\frac {x^2}{a^2}}}{6 a x^2}-\frac {\arcsin \left (\frac {x}{a}\right )}{3 x^3}-\frac {\text {arctanh}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{6 a^3} \]
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {a^2 x \sqrt {1-\frac {x^2}{a^2}}+2 a^3 \csc ^{-1}\left (\frac {a}{x}\right )-x^3 \log (x)+x^3 \log \left (1+\sqrt {1-\frac {x^2}{a^2}}\right )}{6 a^3 x^3} \]
-1/6*(a^2*x*Sqrt[1 - x^2/a^2] + 2*a^3*ArcCsc[a/x] - x^3*Log[x] + x^3*Log[1 + Sqrt[1 - x^2/a^2]])/(a^3*x^3)
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5788, 5138, 243, 52, 73, 221}
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.
\(\displaystyle \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 5788 |
\(\displaystyle \int \frac {\arcsin \left (\frac {x}{a}\right )}{x^4}dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {\int \frac {1}{x^3 \sqrt {1-\frac {x^2}{a^2}}}dx}{3 a}-\frac {\arcsin \left (\frac {x}{a}\right )}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \frac {1}{x^4 \sqrt {1-\frac {x^2}{a^2}}}dx^2}{6 a}-\frac {\arcsin \left (\frac {x}{a}\right )}{3 x^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}}dx^2}{2 a^2}-\frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2}}{6 a}-\frac {\arcsin \left (\frac {x}{a}\right )}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\int \frac {1}{a^2-a^2 x^4}d\sqrt {1-\frac {x^2}{a^2}}-\frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2}}{6 a}-\frac {\arcsin \left (\frac {x}{a}\right )}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {\text {arctanh}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a^2}-\frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2}}{6 a}-\frac {\arcsin \left (\frac {x}{a}\right )}{3 x^3}\) |
3.1.15.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[ArcCsc[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[ u*ArcSin[a/c + b*(x^n/c)]^m, x] /; FreeQ[{a, b, c, n, m}, x]
Time = 1.46 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90
method | result | size |
parts | \(-\frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{3 x^{3}}+\frac {-\frac {\sqrt {1-\frac {x^{2}}{a^{2}}}}{2 x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\frac {x^{2}}{a^{2}}}}\right )}{2 a^{2}}}{3 a}\) | \(54\) |
derivativedivides | \(-\frac {\frac {a^{3} \operatorname {arccsc}\left (\frac {a}{x}\right )}{3 x^{3}}+\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (\frac {a \sqrt {\frac {a^{2}}{x^{2}}-1}}{x}+\ln \left (\frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}-1}\right )\right ) x}{6 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a}}{a^{3}}\) | \(91\) |
default | \(-\frac {\frac {a^{3} \operatorname {arccsc}\left (\frac {a}{x}\right )}{3 x^{3}}+\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (\frac {a \sqrt {\frac {a^{2}}{x^{2}}-1}}{x}+\ln \left (\frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}-1}\right )\right ) x}{6 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a}}{a^{3}}\) | \(91\) |
-1/3*arccsc(a/x)/x^3+1/3/a*(-1/2/x^2*(1-x^2/a^2)^(1/2)-1/2/a^2*arctanh(1/( 1-x^2/a^2)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {4 \, a^{3} \operatorname {arccsc}\left (\frac {a}{x}\right ) + x^{3} \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} + a\right ) - x^{3} \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - a\right ) + 2 \, a x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{12 \, a^{3} x^{3}} \]
-1/12*(4*a^3*arccsc(a/x) + x^3*log(x*sqrt((a^2 - x^2)/x^2) + a) - x^3*log( x*sqrt((a^2 - x^2)/x^2) - a) + 2*a*x^2*sqrt((a^2 - x^2)/x^2))/(a^3*x^3)
Time = 1.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.65 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=- \frac {\operatorname {acsc}{\left (\frac {a}{x} \right )}}{3 x^{3}} + \frac {\begin {cases} - \frac {\sqrt {\frac {a^{2}}{x^{2}} - 1}}{2 a x} - \frac {\operatorname {acosh}{\left (\frac {a}{x} \right )}}{2 a^{2}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\\frac {i a}{2 x^{3} \sqrt {- \frac {a^{2}}{x^{2}} + 1}} - \frac {i}{2 a x \sqrt {- \frac {a^{2}}{x^{2}} + 1}} + \frac {i \operatorname {asin}{\left (\frac {a}{x} \right )}}{2 a^{2}} & \text {otherwise} \end {cases}}{3 a} \]
-acsc(a/x)/(3*x**3) + Piecewise((-sqrt(a**2/x**2 - 1)/(2*a*x) - acosh(a/x) /(2*a**2), Abs(a**2/x**2) > 1), (I*a/(2*x**3*sqrt(-a**2/x**2 + 1)) - I/(2* a*x*sqrt(-a**2/x**2 + 1)) + I*asin(a/x)/(2*a**2), True))/(3*a)
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {\frac {\log \left (\frac {2 \, \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a^{2}} + \frac {\sqrt {-\frac {x^{2}}{a^{2}} + 1}}{x^{2}}}{6 \, a} - \frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{3 \, x^{3}} \]
-1/6*(log(2*sqrt(-x^2/a^2 + 1)/abs(x) + 2/abs(x))/a^2 + sqrt(-x^2/a^2 + 1) /x^2)/a - 1/3*arccsc(a/x)/x^3
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.33 \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=-\frac {a {\left (\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a^{3}} - \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a^{3}} + \frac {2 \, \sqrt {a^{2} - x^{2}}}{a^{2} x^{2}}\right )}}{12 \, {\left | a \right |}} - \frac {\arcsin \left (\frac {x}{a}\right )}{3 \, x^{3}} \]
-1/12*a*(log(abs(a + sqrt(a^2 - x^2)))/a^3 - log(abs(-a + sqrt(a^2 - x^2)) )/a^3 + 2*sqrt(a^2 - x^2)/(a^2*x^2))/abs(a) - 1/3*arcsin(x/a)/x^3
Timed out. \[ \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^4} \, dx=\int \frac {\mathrm {asin}\left (\frac {x}{a}\right )}{x^4} \,d x \]