Optimal. Leaf size=68 \[ \frac {\sqrt {-1+x}}{18 x^3}+\frac {5 \sqrt {-1+x}}{72 x^2}+\frac {5 \sqrt {-1+x}}{48 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5}{48} \text {ArcTan}\left (\sqrt {-1+x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5378, 12, 44,
65, 209} \begin {gather*} \frac {5}{48} \text {ArcTan}\left (\sqrt {x-1}\right )+\frac {\sqrt {x-1}}{18 x^3}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5 \sqrt {x-1}}{72 x^2}+\frac {5 \sqrt {x-1}}{48 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 65
Rule 209
Rule 5378
Rubi steps
\begin {align*} \int \frac {\sec ^{-1}\left (\sqrt {x}\right )}{x^4} \, dx &=-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{3} \int \frac {1}{2 \sqrt {-1+x} x^4} \, dx\\ &=-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {1}{6} \int \frac {1}{\sqrt {-1+x} x^4} \, dx\\ &=\frac {\sqrt {-1+x}}{18 x^3}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5}{36} \int \frac {1}{\sqrt {-1+x} x^3} \, dx\\ &=\frac {\sqrt {-1+x}}{18 x^3}+\frac {5 \sqrt {-1+x}}{72 x^2}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5}{48} \int \frac {1}{\sqrt {-1+x} x^2} \, dx\\ &=\frac {\sqrt {-1+x}}{18 x^3}+\frac {5 \sqrt {-1+x}}{72 x^2}+\frac {5 \sqrt {-1+x}}{48 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5}{96} \int \frac {1}{\sqrt {-1+x} x} \, dx\\ &=\frac {\sqrt {-1+x}}{18 x^3}+\frac {5 \sqrt {-1+x}}{72 x^2}+\frac {5 \sqrt {-1+x}}{48 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5}{48} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {\sqrt {-1+x}}{18 x^3}+\frac {5 \sqrt {-1+x}}{72 x^2}+\frac {5 \sqrt {-1+x}}{48 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5}{48} \tan ^{-1}\left (\sqrt {-1+x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 45, normalized size = 0.66 \begin {gather*} \frac {\sqrt {-1+x} \left (8+10 x+15 x^2\right )-48 \sec ^{-1}\left (\sqrt {x}\right )-15 x^3 \text {ArcSin}\left (\frac {1}{\sqrt {x}}\right )}{144 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 67, normalized size = 0.99
method | result | size |
derivativedivides | \(-\frac {\mathrm {arcsec}\left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {x -1}\, \left (15 \arctan \left (\frac {1}{\sqrt {x -1}}\right ) x^{3}-15 \sqrt {x -1}\, x^{2}-10 \sqrt {x -1}\, x -8 \sqrt {x -1}\right )}{144 \sqrt {\frac {x -1}{x}}\, x^{\frac {7}{2}}}\) | \(67\) |
default | \(-\frac {\mathrm {arcsec}\left (\sqrt {x}\right )}{3 x^{3}}-\frac {\sqrt {x -1}\, \left (15 \arctan \left (\frac {1}{\sqrt {x -1}}\right ) x^{3}-15 \sqrt {x -1}\, x^{2}-10 \sqrt {x -1}\, x -8 \sqrt {x -1}\right )}{144 \sqrt {\frac {x -1}{x}}\, x^{\frac {7}{2}}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs.
\(2 (48) = 96\).
time = 0.47, size = 106, normalized size = 1.56 \begin {gather*} -\frac {15 \, x^{\frac {5}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{2}} + 40 \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x} \sqrt {-\frac {1}{x} + 1}}{144 \, {\left (x^{3} {\left (\frac {1}{x} - 1\right )}^{3} - 3 \, x^{2} {\left (\frac {1}{x} - 1\right )}^{2} + 3 \, x {\left (\frac {1}{x} - 1\right )} - 1\right )}} - \frac {\operatorname {arcsec}\left (\sqrt {x}\right )}{3 \, x^{3}} + \frac {5}{48} \, \arctan \left (\sqrt {x} \sqrt {-\frac {1}{x} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 5.12, size = 35, normalized size = 0.51 \begin {gather*} \frac {3 \, {\left (5 \, x^{3} - 16\right )} \operatorname {arcsec}\left (\sqrt {x}\right ) + {\left (15 \, x^{2} + 10 \, x + 8\right )} \sqrt {x - 1}}{144 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 138.71, size = 180, normalized size = 2.65 \begin {gather*} \frac {\begin {cases} \frac {5 i \operatorname {acosh}{\left (\frac {1}{\sqrt {x}} \right )}}{8} - \frac {5 i}{8 \sqrt {x} \sqrt {-1 + \frac {1}{x}}} + \frac {5 i}{24 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{12 x^{\frac {5}{2}} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{3 x^{\frac {7}{2}} \sqrt {-1 + \frac {1}{x}}} & \text {for}\: \frac {1}{\left |{x}\right |} > 1 \\- \frac {5 \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{8} + \frac {5}{8 \sqrt {x} \sqrt {1 - \frac {1}{x}}} - \frac {5}{24 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x}}} - \frac {1}{12 x^{\frac {5}{2}} \sqrt {1 - \frac {1}{x}}} - \frac {1}{3 x^{\frac {7}{2}} \sqrt {1 - \frac {1}{x}}} & \text {otherwise} \end {cases}}{6} - \frac {\operatorname {asec}{\left (\sqrt {x} \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.53, size = 58, normalized size = 0.85 \begin {gather*} \frac {5 \, \sqrt {-\frac {1}{x} + 1}}{48 \, \sqrt {x}} + \frac {5 \, \sqrt {-\frac {1}{x} + 1}}{72 \, x^{\frac {3}{2}}} + \frac {\sqrt {-\frac {1}{x} + 1}}{18 \, x^{\frac {5}{2}}} - \frac {\arccos \left (\frac {1}{\sqrt {x}}\right )}{3 \, x^{3}} + \frac {5}{48} \, \arccos \left (\frac {1}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {acos}\left (\frac {1}{\sqrt {x}}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________