Optimal. Leaf size=59 \[ -\frac {1}{18 x^{3/2}}+\frac {1}{30 x^{5/2}}-\frac {\pi }{12 x^3}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}+\frac {1}{6 \sqrt {x}}+\frac {1}{6} \tan ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5159, 30, 5033, 51, 63, 203} \[ -\frac {1}{18 x^{3/2}}+\frac {1}{30 x^{5/2}}-\frac {\pi }{12 x^3}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}+\frac {1}{6 \sqrt {x}}+\frac {1}{6} \tan ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 51
Rule 63
Rule 203
Rule 5033
Rule 5159
Rubi steps
\begin {align*} \int -\frac {\tan ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right )}{x^4} \, dx &=-\left (\frac {1}{2} \int \frac {\tan ^{-1}\left (\sqrt {x}\right )}{x^4} \, dx\right )+\frac {1}{4} \pi \int \frac {1}{x^4} \, dx\\ &=-\frac {\pi }{12 x^3}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}-\frac {1}{12} \int \frac {1}{x^{7/2} (1+x)} \, dx\\ &=-\frac {\pi }{12 x^3}+\frac {1}{30 x^{5/2}}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}+\frac {1}{12} \int \frac {1}{x^{5/2} (1+x)} \, dx\\ &=-\frac {\pi }{12 x^3}+\frac {1}{30 x^{5/2}}-\frac {1}{18 x^{3/2}}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}-\frac {1}{12} \int \frac {1}{x^{3/2} (1+x)} \, dx\\ &=-\frac {\pi }{12 x^3}+\frac {1}{30 x^{5/2}}-\frac {1}{18 x^{3/2}}+\frac {1}{6 \sqrt {x}}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}+\frac {1}{12} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=-\frac {\pi }{12 x^3}+\frac {1}{30 x^{5/2}}-\frac {1}{18 x^{3/2}}+\frac {1}{6 \sqrt {x}}+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\pi }{12 x^3}+\frac {1}{30 x^{5/2}}-\frac {1}{18 x^{3/2}}+\frac {1}{6 \sqrt {x}}+\frac {1}{6} \tan ^{-1}\left (\sqrt {x}\right )+\frac {\tan ^{-1}\left (\sqrt {x}\right )}{6 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 51, normalized size = 0.86 \[ \frac {1}{90} \left (\frac {30 \tan ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )}{x^3}-\frac {-15 x^2+5 x-3}{x^{5/2}}+15 \tan ^{-1}\left (\sqrt {x}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 40, normalized size = 0.68 \[ -\frac {30 \, {\left (x^{3} + 1\right )} \arctan \left (\sqrt {x + 1} - \sqrt {x}\right ) - {\left (15 \, x^{2} - 5 \, x + 3\right )} \sqrt {x}}{90 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 39, normalized size = 0.66 \[ \frac {15 \, x^{2} - 5 \, x + 3}{90 \, x^{\frac {5}{2}}} + \frac {\arctan \left (-\sqrt {x + 1} + \sqrt {x}\right )}{3 \, x^{3}} + \frac {1}{6} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 40, normalized size = 0.68 \[ \frac {\arctan \left (\sqrt {x}-\sqrt {x +1}\right )}{3 x^{3}}+\frac {1}{30 x^{\frac {5}{2}}}-\frac {1}{18 x^{\frac {3}{2}}}+\frac {1}{6 \sqrt {x}}+\frac {\arctan \left (\sqrt {x}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 39, normalized size = 0.66 \[ \frac {1}{6 \, \sqrt {x}} - \frac {1}{18 \, x^{\frac {3}{2}}} - \frac {\arctan \left (\sqrt {x + 1} - \sqrt {x}\right )}{3 \, x^{3}} + \frac {1}{30 \, x^{\frac {5}{2}}} + \frac {1}{6} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.94, size = 56, normalized size = 0.95 \[ -\frac {\frac {\mathrm {atan}\left (\sqrt {x+1}-\sqrt {x}\right )}{3}-\frac {\sqrt {x}}{30}+\frac {x^{3/2}}{18}-\frac {x^{5/2}}{6}}{x^3}+\frac {\ln \left (\frac {{\left (-1+\sqrt {x}\,1{}\mathrm {i}\right )}^2}{x+1}\right )\,1{}\mathrm {i}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________