Optimal. Leaf size=110 \[ \frac{2 \left (1-21 x^2\right )}{27 \left (x^2\right )^{3/2}}+\frac{\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^3}{3 x^3}+\frac{\left (x^2-1\right ) \sec ^{-1}(x)^2}{3 \left (x^2\right )^{3/2}}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}-\frac{2 \left (x^2-1\right )^{3/2} \sec ^{-1}(x)}{9 x^3}-\frac{4 \sqrt{x^2-1} \sec ^{-1}(x)}{3 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.199997, antiderivative size = 146, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5242, 4678, 4650, 4620, 8} \[ \frac{2 \sqrt{x^2}}{27 x^4}-\frac{14}{9 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}+\frac{\left (1-\frac{1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}-\frac{2 \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)}{9 x}-\frac{4 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \sec ^{-1}(x)}{3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5242
Rule 4678
Rule 4650
Rule 4620
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx &=-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int x \sqrt{1-x^2} \cos ^{-1}(x)^3 \, dx,x,\frac{1}{x}\right )}{x}\\ &=\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}+\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \left (1-x^2\right ) \cos ^{-1}(x)^2 \, dx,x,\frac{1}{x}\right )}{x}\\ &=\frac{\left (1-\frac{1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}+\frac{\left (2 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int x \sqrt{1-x^2} \cos ^{-1}(x) \, dx,x,\frac{1}{x}\right )}{3 x}+\frac{\left (2 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \cos ^{-1}(x)^2 \, dx,x,\frac{1}{x}\right )}{3 x}\\ &=-\frac{2 \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)}{9 x}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}-\frac{\left (2 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\frac{1}{x}\right )}{9 x}+\frac{\left (4 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{x \cos ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,\frac{1}{x}\right )}{3 x}\\ &=-\frac{2}{9 \sqrt{x^2}}+\frac{2 \sqrt{x^2}}{27 x^4}-\frac{4 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \sec ^{-1}(x)}{3 x}-\frac{2 \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)}{9 x}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}-\frac{\left (4 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\frac{1}{x}\right )}{3 x}\\ &=-\frac{14}{9 \sqrt{x^2}}+\frac{2 \sqrt{x^2}}{27 x^4}-\frac{4 \sqrt{1-\frac{1}{x^2}} \sqrt{x^2} \sec ^{-1}(x)}{3 x}-\frac{2 \left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)}{9 x}+\frac{2 \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sqrt{x^2} \sec ^{-1}(x)^3}{3 x}\\ \end{align*}
Mathematica [A] time = 0.0896488, size = 92, normalized size = 0.84 \[ \frac{2 \sqrt{1-\frac{1}{x^2}} x \left (1-21 x^2\right )+9 \left (x^2-1\right )^2 \sec ^{-1}(x)^3+9 \sqrt{1-\frac{1}{x^2}} x \left (3 x^2-1\right ) \sec ^{-1}(x)^2-6 \left (7 x^4-8 x^2+1\right ) \sec ^{-1}(x)}{27 x^3 \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.588, size = 1153, normalized size = 10.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.53848, size = 126, normalized size = 1.15 \begin{align*} \frac{{\left (x^{2} - 1\right )}^{\frac{3}{2}} \operatorname{arcsec}\left (x\right )^{3}}{3 \, x^{3}} + \frac{{\left (3 \, x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )^{2}}{3 \, x^{3}} - \frac{2 \,{\left ({\left (21 \, x^{2} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} + 3 \,{\left (7 \, x^{4} - 8 \, x^{2} + 1\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1}\right )\right )}}{27 \, \sqrt{x + 1} \sqrt{x - 1} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.43214, size = 163, normalized size = 1.48 \begin{align*} \frac{9 \,{\left (3 \, x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )^{2} - 42 \, x^{2} + 3 \,{\left (3 \,{\left (x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )^{3} - 2 \,{\left (7 \, x^{2} - 1\right )} \operatorname{arcsec}\left (x\right )\right )} \sqrt{x^{2} - 1} + 2}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right )^{3}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]