Optimal. Leaf size=68 \[ \frac {x^{7/2}}{56}-\frac {x^{5/2}}{40}+\frac {x^{3/2}}{24}+\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x}}{8}+\frac {1}{8} \tan ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5159, 30, 5033, 50, 63, 203} \[ \frac {\pi x^4}{16}+\frac {x^{7/2}}{56}-\frac {x^{5/2}}{40}+\frac {x^{3/2}}{24}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x}}{8}+\frac {1}{8} \tan ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 30
Rule 50
Rule 63
Rule 203
Rule 5033
Rule 5159
Rubi steps
\begin {align*} \int -x^3 \tan ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right ) \, dx &=-\left (\frac {1}{2} \int x^3 \tan ^{-1}\left (\sqrt {x}\right ) \, dx\right )+\frac {1}{4} \pi \int x^3 \, dx\\ &=\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{16} \int \frac {x^{7/2}}{1+x} \, dx\\ &=\frac {x^{7/2}}{56}+\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{16} \int \frac {x^{5/2}}{1+x} \, dx\\ &=-\frac {x^{5/2}}{40}+\frac {x^{7/2}}{56}+\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{16} \int \frac {x^{3/2}}{1+x} \, dx\\ &=\frac {x^{3/2}}{24}-\frac {x^{5/2}}{40}+\frac {x^{7/2}}{56}+\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{16} \int \frac {\sqrt {x}}{1+x} \, dx\\ &=-\frac {\sqrt {x}}{8}+\frac {x^{3/2}}{24}-\frac {x^{5/2}}{40}+\frac {x^{7/2}}{56}+\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{16} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=-\frac {\sqrt {x}}{8}+\frac {x^{3/2}}{24}-\frac {x^{5/2}}{40}+\frac {x^{7/2}}{56}+\frac {\pi x^4}{16}-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{8}+\frac {x^{3/2}}{24}-\frac {x^{5/2}}{40}+\frac {x^{7/2}}{56}+\frac {\pi x^4}{16}+\frac {1}{8} \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{8} x^4 \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 58, normalized size = 0.85 \[ \frac {1}{8} \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{840} \sqrt {x} \left (210 x^{7/2} \tan ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )-15 x^3+21 x^2-35 x+105\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 40, normalized size = 0.59 \[ \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (\sqrt {x + 1} - \sqrt {x}\right ) + \frac {1}{840} \, {\left (15 \, x^{3} - 21 \, x^{2} + 35 \, x - 105\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 44, normalized size = 0.65 \[ -\frac {1}{4} \, x^{4} \arctan \left (-\sqrt {x + 1} + \sqrt {x}\right ) + \frac {1}{56} \, x^{\frac {7}{2}} - \frac {1}{40} \, x^{\frac {5}{2}} + \frac {1}{24} \, x^{\frac {3}{2}} - \frac {1}{8} \, \sqrt {x} + \frac {1}{8} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 0.66 \[ -\frac {x^{4} \arctan \left (\sqrt {x}-\sqrt {x +1}\right )}{4}+\frac {x^{\frac {7}{2}}}{56}-\frac {x^{\frac {5}{2}}}{40}+\frac {x^{\frac {3}{2}}}{24}-\frac {\sqrt {x}}{8}+\frac {\arctan \left (\sqrt {x}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 44, normalized size = 0.65 \[ \frac {1}{4} \, x^{4} \arctan \left (\sqrt {x + 1} - \sqrt {x}\right ) + \frac {1}{56} \, x^{\frac {7}{2}} - \frac {1}{40} \, x^{\frac {5}{2}} + \frac {1}{24} \, x^{\frac {3}{2}} - \frac {1}{8} \, \sqrt {x} + \frac {1}{8} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 72, normalized size = 1.06 \[ \frac {x^{3/2}}{24}-\frac {\sqrt {x}}{8}-\frac {x^{5/2}}{40}+\frac {x^{7/2}}{56}+\frac {\mathrm {atan}\left (\sqrt {x+1}-\sqrt {x}\right )\,\left (\frac {x^5}{2}+\frac {x^4}{2}\right )}{2\,x+2}+\frac {\ln \left (\frac {{\left (-1+\sqrt {x}\,1{}\mathrm {i}\right )}^2}{x+1}\right )\,1{}\mathrm {i}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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