Integrand size = 17, antiderivative size = 74 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {24 \sqrt {-1+x^2}}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}}-\frac {12 \sqrt {-1+x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {-1+x^2} \csc ^{-1}(x)^4}{x} \]
24*arccsc(x)/(x^2)^(1/2)-4*arccsc(x)^3/(x^2)^(1/2)+24*(x^2-1)^(1/2)/x-12*a rccsc(x)^2*(x^2-1)^(1/2)/x+arccsc(x)^4*(x^2-1)^(1/2)/x
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {24 \left (-1+x^2\right )+24 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)-12 \left (-1+x^2\right ) \csc ^{-1}(x)^2-4 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)^3+\left (-1+x^2\right ) \csc ^{-1}(x)^4}{x \sqrt {-1+x^2}} \]
(24*(-1 + x^2) + 24*Sqrt[1 - x^(-2)]*x*ArcCsc[x] - 12*(-1 + x^2)*ArcCsc[x] ^2 - 4*Sqrt[1 - x^(-2)]*x*ArcCsc[x]^3 + (-1 + x^2)*ArcCsc[x]^4)/(x*Sqrt[-1 + x^2])
Time = 0.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5766, 5182, 5130, 5182, 5130, 241}
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.
\(\displaystyle \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {x^2-1}} \, dx\) |
\(\Big \downarrow \) 5766 |
\(\displaystyle -\frac {\sqrt {x^2} \int \frac {\arcsin \left (\frac {1}{x}\right )^4}{\sqrt {1-\frac {1}{x^2}} x}d\frac {1}{x}}{x}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle -\frac {\sqrt {x^2} \left (4 \int \arcsin \left (\frac {1}{x}\right )^3d\frac {1}{x}-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^4\right )}{x}\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle -\frac {\sqrt {x^2} \left (4 \left (\frac {\arcsin \left (\frac {1}{x}\right )^3}{x}-3 \int \frac {\arcsin \left (\frac {1}{x}\right )^2}{\sqrt {1-\frac {1}{x^2}} x}d\frac {1}{x}\right )-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^4\right )}{x}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle -\frac {\sqrt {x^2} \left (4 \left (\frac {\arcsin \left (\frac {1}{x}\right )^3}{x}-3 \left (2 \int \arcsin \left (\frac {1}{x}\right )d\frac {1}{x}-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^2\right )\right )-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^4\right )}{x}\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle -\frac {\sqrt {x^2} \left (4 \left (\frac {\arcsin \left (\frac {1}{x}\right )^3}{x}-3 \left (2 \left (\frac {\arcsin \left (\frac {1}{x}\right )}{x}-\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x}d\frac {1}{x}\right )-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^2\right )\right )-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^4\right )}{x}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle -\frac {\sqrt {x^2} \left (4 \left (\frac {\arcsin \left (\frac {1}{x}\right )^3}{x}-3 \left (2 \left (\frac {\arcsin \left (\frac {1}{x}\right )}{x}+\sqrt {1-\frac {1}{x^2}}\right )-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^2\right )\right )-\sqrt {1-\frac {1}{x^2}} \arcsin \left (\frac {1}{x}\right )^4\right )}{x}\) |
-((Sqrt[x^2]*(-(Sqrt[1 - x^(-2)]*ArcSin[x^(-1)]^4) + 4*(ArcSin[x^(-1)]^3/x - 3*(-(Sqrt[1 - x^(-2)]*ArcSin[x^(-1)]^2) + 2*(Sqrt[1 - x^(-2)] + ArcSin[ x^(-1)]/x)))))/x)
3.7.92.3.1 Defintions of rubi rules used
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) ^2)^(p_), x_Symbol] :> Simp[-Sqrt[x^2]/x Subst[Int[(e + d*x^2)^p*((a + b* ArcSin[x/c])^n/x^(m + 2*(p + 1))), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1 /2] && GtQ[e, 0] && LtQ[d, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, \left (\operatorname {arccsc}\left (x \right )^{4} x^{2}-\operatorname {arccsc}\left (x \right )^{4}-12 \operatorname {arccsc}\left (x \right )^{2} x^{2}+12 \operatorname {arccsc}\left (x \right )^{2}-4 \sqrt {\frac {x^{2}-1}{x^{2}}}\, \operatorname {arccsc}\left (x \right )^{3} x +24 x^{2}-24+24 \,\operatorname {arccsc}\left (x \right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right )}{x^{2}-1}\) | \(102\) |
csgn(x*(1-1/x^2)^(1/2))*((x^2-1)/x^2)^(1/2)/(x^2-1)*(arccsc(x)^4*x^2-arccs c(x)^4-12*arccsc(x)^2*x^2+12*arccsc(x)^2-4*((x^2-1)/x^2)^(1/2)*arccsc(x)^3 *x+24*x^2-24+24*arccsc(x)*((x^2-1)/x^2)^(1/2)*x)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.50 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=-\frac {4 \, \operatorname {arccsc}\left (x\right )^{3} - {\left (\operatorname {arccsc}\left (x\right )^{4} - 12 \, \operatorname {arccsc}\left (x\right )^{2} + 24\right )} \sqrt {x^{2} - 1} - 24 \, \operatorname {arccsc}\left (x\right )}{x} \]
\[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {\operatorname {acsc}^{4}{\left (x \right )}}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\frac {\sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right )^{4}}{x} - 12 \, \sqrt {-\frac {1}{x^{2}} + 1} \operatorname {arccsc}\left (x\right )^{2} - \frac {4 \, \operatorname {arccsc}\left (x\right )^{3}}{x} + 24 \, \sqrt {-\frac {1}{x^{2}} + 1} + \frac {24 \, \operatorname {arccsc}\left (x\right )}{x} \]
sqrt(x^2 - 1)*arccsc(x)^4/x - 12*sqrt(-1/x^2 + 1)*arccsc(x)^2 - 4*arccsc(x )^3/x + 24*sqrt(-1/x^2 + 1) + 24*arccsc(x)/x
\[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\int { \frac {\operatorname {arccsc}\left (x\right )^{4}}{\sqrt {x^{2} - 1} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx=\int \frac {{\mathrm {asin}\left (\frac {1}{x}\right )}^4}{x^2\,\sqrt {x^2-1}} \,d x \]