Integrand size = 12, antiderivative size = 60 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {a+b \sec ^{-1}(c x)}{3 x^3} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5328, 272, 45} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=-\frac {a+b \sec ^{-1}(c x)}{3 x^3}-\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \]
[In]
[Out]
Rule 45
Rule 272
Rule 5328
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{3 x^3}+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^5} \, dx}{3 c} \\ & = -\frac {a+b \sec ^{-1}(c x)}{3 x^3}-\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{6 c} \\ & = -\frac {a+b \sec ^{-1}(c x)}{3 x^3}-\frac {b \text {Subst}\left (\int \left (\frac {c^2}{\sqrt {1-\frac {x}{c^2}}}-c^2 \sqrt {1-\frac {x}{c^2}}\right ) \, dx,x,\frac {1}{x^2}\right )}{6 c} \\ & = \frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {a+b \sec ^{-1}(c x)}{3 x^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=-\frac {a}{3 x^3}+b \left (\frac {2 c^3}{9}+\frac {c}{9 x^2}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \sec ^{-1}(c x)}{3 x^3} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18
method | result | size |
parts | \(-\frac {a}{3 x^{3}}+b \,c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\) | \(71\) |
derivativedivides | \(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(75\) |
default | \(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(75\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.67 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=-\frac {3 \, b \operatorname {arcsec}\left (c x\right ) - {\left (2 \, b c^{2} x^{2} + b\right )} \sqrt {c^{2} x^{2} - 1} + 3 \, a}{9 \, x^{3}} \]
[In]
[Out]
Time = 1.51 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.83 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=- \frac {a}{3 x^{3}} - \frac {b \operatorname {asec}{\left (c x \right )}}{3 x^{3}} + \frac {b \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=-\frac {1}{9} \, b {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {3 \, \operatorname {arcsec}\left (c x\right )}{x^{3}}\right )} - \frac {a}{3 \, x^{3}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=\frac {1}{9} \, {\left (2 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} - \frac {3 \, b \arccos \left (\frac {1}{c x}\right )}{c x^{3}} - \frac {3 \, a}{c x^{3}}\right )} c \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^4} \,d x \]
[In]
[Out]