Optimal. Leaf size=104 \[ -128 \sqrt [4]{x}+x-\frac {256 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt [3]{2}-\sqrt [4]{x}}{\sqrt [3]{2} \sqrt {3}}\right )}{\sqrt {3}}+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (4\ 2^{2/3}-2 \sqrt [3]{2} \sqrt [4]{x}+\sqrt {x}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {383, 470, 327,
206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {256 \sqrt [3]{2} \text {ArcTan}\left (\frac {\sqrt [3]{2}-\sqrt [4]{x}}{\sqrt [3]{2} \sqrt {3}}\right )}{\sqrt {3}}+x-128 \sqrt [4]{x}+\frac {256}{3} \sqrt [3]{2} \log \left (\sqrt [4]{x}+2 \sqrt [3]{2}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (\sqrt {x}-2 \sqrt [3]{2} \sqrt [4]{x}+4\ 2^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 327
Rule 383
Rule 470
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {-16+x^{3/4}}{16+x^{3/4}} \, dx &=4 \text {Subst}\left (\int \frac {x^3 \left (-16+x^3\right )}{16+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=x-128 \text {Subst}\left (\int \frac {x^3}{16+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+2048 \text {Subst}\left (\int \frac {1}{16+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+\frac {1}{3} \left (256 \sqrt [3]{2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+x} \, dx,x,\sqrt [4]{x}\right )+\frac {1}{3} \left (256 \sqrt [3]{2}\right ) \text {Subst}\left (\int \frac {4 \sqrt [3]{2}-x}{4\ 2^{2/3}-2 \sqrt [3]{2} x+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {1}{3} \left (128 \sqrt [3]{2}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{2}+2 x}{4\ 2^{2/3}-2 \sqrt [3]{2} x+x^2} \, dx,x,\sqrt [4]{x}\right )+\left (256\ 2^{2/3}\right ) \text {Subst}\left (\int \frac {1}{4\ 2^{2/3}-2 \sqrt [3]{2} x+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (4\ 2^{2/3}-2 \sqrt [3]{2} \sqrt [4]{x}+\sqrt {x}\right )+\left (256 \sqrt [3]{2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {\sqrt [4]{x}}{\sqrt [3]{2}}\right )\\ &=-128 \sqrt [4]{x}+x-\frac {256 \sqrt [3]{2} \tan ^{-1}\left (\frac {2-2^{2/3} \sqrt [4]{x}}{2 \sqrt {3}}\right )}{\sqrt {3}}+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (4\ 2^{2/3}-2 \sqrt [3]{2} \sqrt [4]{x}+\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 104, normalized size = 1.00 \begin {gather*} -128 \sqrt [4]{x}+x-\frac {256 \sqrt [3]{2} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [4]{x}}{\sqrt [3]{2} \sqrt {3}}\right )}{\sqrt {3}}+\frac {256}{3} \sqrt [3]{2} \log \left (4+2^{2/3} \sqrt [4]{x}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (-8+2\ 2^{2/3} \sqrt [4]{x}-\sqrt [3]{2} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 56.06, size = 66, normalized size = 0.63
method | result | size |
derivativedivides | \(x -128 x^{\frac {1}{4}}+\frac {128 \,16^{\frac {1}{3}} \ln \left (x^{\frac {1}{4}}+16^{\frac {1}{3}}\right )}{3}-\frac {64 \,16^{\frac {1}{3}} \ln \left (\sqrt {x}-16^{\frac {1}{3}} x^{\frac {1}{4}}+16^{\frac {2}{3}}\right )}{3}+\frac {128 \,16^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {16^{\frac {2}{3}} x^{\frac {1}{4}}}{8}-1\right )}{3}\right )}{3}\) | \(66\) |
default | \(x -128 x^{\frac {1}{4}}+\frac {128 \,16^{\frac {1}{3}} \ln \left (x^{\frac {1}{4}}+16^{\frac {1}{3}}\right )}{3}-\frac {64 \,16^{\frac {1}{3}} \ln \left (\sqrt {x}-16^{\frac {1}{3}} x^{\frac {1}{4}}+16^{\frac {2}{3}}\right )}{3}+\frac {128 \,16^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {16^{\frac {2}{3}} x^{\frac {1}{4}}}{8}-1\right )}{3}\right )}{3}\) | \(66\) |
meijerg | \(-\frac {128 \,2^{\frac {1}{3}} \left (\frac {3 x^{\frac {1}{4}} 2^{\frac {2}{3}}}{4}-\frac {x^{\frac {1}{4}} 2^{\frac {2}{3}} \left (\frac {2 \,2^{\frac {1}{3}} \ln \left (1+\frac {x^{\frac {1}{4}} 2^{\frac {2}{3}}}{4}\right )}{x^{\frac {1}{4}}}-\frac {2^{\frac {1}{3}} \ln \left (1-\frac {x^{\frac {1}{4}} 2^{\frac {2}{3}}}{4}+\frac {2^{\frac {1}{3}} \sqrt {x}}{8}\right )}{x^{\frac {1}{4}}}+\frac {2 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {2}{3}} x^{\frac {1}{4}}}{8-x^{\frac {1}{4}} 2^{\frac {2}{3}}}\right )}{x^{\frac {1}{4}}}\right )}{4}\right )}{3}+\frac {128 \,2^{\frac {1}{3}} \left (-\frac {3 x^{\frac {1}{4}} 2^{\frac {2}{3}} \left (-\frac {7 x^{\frac {3}{4}}}{16}+28\right )}{112}+\frac {x^{\frac {1}{4}} 2^{\frac {2}{3}} \left (\frac {2 \,2^{\frac {1}{3}} \ln \left (1+\frac {x^{\frac {1}{4}} 2^{\frac {2}{3}}}{4}\right )}{x^{\frac {1}{4}}}-\frac {2^{\frac {1}{3}} \ln \left (1-\frac {x^{\frac {1}{4}} 2^{\frac {2}{3}}}{4}+\frac {2^{\frac {1}{3}} \sqrt {x}}{8}\right )}{x^{\frac {1}{4}}}+\frac {2 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {2}{3}} x^{\frac {1}{4}}}{8-x^{\frac {1}{4}} 2^{\frac {2}{3}}}\right )}{x^{\frac {1}{4}}}\right )}{4}\right )}{3}\) | \(217\) |
trager | \(\text {Expression too large to display}\) | \(2796\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 71, normalized size = 0.68 \begin {gather*} \frac {256}{3} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} - x^{\frac {1}{4}}\right )}\right ) - \frac {128}{3} \cdot 2^{\frac {1}{3}} \log \left (4 \cdot 2^{\frac {2}{3}} - 2 \cdot 2^{\frac {1}{3}} x^{\frac {1}{4}} + \sqrt {x}\right ) + \frac {256}{3} \cdot 2^{\frac {1}{3}} \log \left (2 \cdot 2^{\frac {1}{3}} + x^{\frac {1}{4}}\right ) + x - 128 \, x^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 71, normalized size = 0.68 \begin {gather*} \frac {256}{3} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} x^{\frac {1}{4}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {128}{3} \cdot 2^{\frac {1}{3}} \log \left (4 \cdot 2^{\frac {2}{3}} - 2 \cdot 2^{\frac {1}{3}} x^{\frac {1}{4}} + \sqrt {x}\right ) + \frac {256}{3} \cdot 2^{\frac {1}{3}} \log \left (2 \cdot 2^{\frac {1}{3}} + x^{\frac {1}{4}}\right ) + x - 128 \, x^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.85, size = 102, normalized size = 0.98 \begin {gather*} - 128 \sqrt [4]{x} + x + \frac {256 \cdot \sqrt [3]{2} \log {\left (\sqrt [4]{x} + 2 \cdot \sqrt [3]{2} \right )}}{3} - \frac {128 \cdot \sqrt [3]{2} \log {\left (- 2 \cdot \sqrt [3]{2} \sqrt [4]{x} + \sqrt {x} + 4 \cdot 2^{\frac {2}{3}} \right )}}{3} + \frac {256 \cdot \sqrt [3]{2} \sqrt {3} \operatorname {atan}{\left (\frac {2^{\frac {2}{3}} \sqrt {3} \sqrt [4]{x}}{6} - \frac {\sqrt {3}}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 71, normalized size = 0.68 \begin {gather*} \frac {256}{3} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} - x^{\frac {1}{4}}\right )}\right ) - \frac {128}{3} \cdot 2^{\frac {1}{3}} \log \left (4 \cdot 2^{\frac {2}{3}} - 2 \cdot 2^{\frac {1}{3}} x^{\frac {1}{4}} + \sqrt {x}\right ) + \frac {256}{3} \cdot 2^{\frac {1}{3}} \log \left (2 \cdot 2^{\frac {1}{3}} + x^{\frac {1}{4}}\right ) + x - 128 \, x^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.50, size = 90, normalized size = 0.87 \begin {gather*} x+\frac {256\,2^{1/3}\,\ln \left (12288\,2^{1/3}+6144\,x^{1/4}\right )}{3}-128\,x^{1/4}+\frac {128\,2^{1/3}\,\ln \left (6144\,x^{1/4}+6144\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3}-\frac {128\,2^{1/3}\,\ln \left (6144\,x^{1/4}-6144\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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