Optimal. Leaf size=157 \[ -\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2} \sqrt [3]{d} x}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\tan ^{-1}\left (\sqrt {-1+d x^3}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {-1+d x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {498}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{2} \sqrt [3]{d} x+1}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\text {ArcTan}\left (\sqrt {d x^3-1}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {d x^3-1}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d x^3-1}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 498
Rubi steps
\begin {align*} \int \frac {x}{\left (4-d x^3\right ) \sqrt {-1+d x^3}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2} \sqrt [3]{d} x}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\tan ^{-1}\left (\sqrt {-1+d x^3}\right )}{9\ 2^{2/3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {-1+d x^3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {-1+d x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3} d^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 10.04, size = 54, normalized size = 0.34 \begin {gather*} \frac {x^2 \sqrt {1-d x^3} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};d x^3,\frac {d x^3}{4}\right )}{8 \sqrt {-1+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.23, size = 240, normalized size = 1.53
method | result | size |
default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-4\right )}{\sum }\frac {\sqrt {-\frac {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}{2}}\, \sqrt {\frac {x -\frac {1}{d^{\frac {1}{3}}}}{-\frac {3}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}}}\, \sqrt {2}\, \sqrt {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} d +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}-i \sqrt {3}\, d^{\frac {1}{3}}+\underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}+d^{\frac {1}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-i \left (x +\frac {1}{2 d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right ) \sqrt {3}\, d^{\frac {1}{3}}}}{3}, \frac {i \sqrt {3}\, d^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}-\frac {i \sqrt {3}\, d^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha }{6}-\frac {i \sqrt {3}}{6}+\frac {d^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {1}{2}, \sqrt {-\frac {i \sqrt {3}}{d^{\frac {1}{3}} \left (-\frac {3}{2 d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {4}{3}} \sqrt {d \,x^{3}-1}}\right )}{9}\) | \(240\) |
elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-4\right )}{\sum }\frac {\sqrt {-\frac {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}{2}}\, \sqrt {\frac {x -\frac {1}{d^{\frac {1}{3}}}}{-\frac {3}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}}}\, \sqrt {2}\, \sqrt {i \left (2 x +\frac {1}{d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{d^{\frac {1}{3}}}\right ) d^{\frac {1}{3}}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} d +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}-i \sqrt {3}\, d^{\frac {1}{3}}+\underline {\hspace {1.25 ex}}\alpha \,d^{\frac {2}{3}}+d^{\frac {1}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-i \left (x +\frac {1}{2 d^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right ) \sqrt {3}\, d^{\frac {1}{3}}}}{3}, \frac {i \sqrt {3}\, d^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}-\frac {i \sqrt {3}\, d^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha }{6}-\frac {i \sqrt {3}}{6}+\frac {d^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha }{2}-\frac {1}{2}, \sqrt {-\frac {i \sqrt {3}}{d^{\frac {1}{3}} \left (-\frac {3}{2 d^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 d^{\frac {1}{3}}}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \,d^{\frac {4}{3}} \sqrt {d \,x^{3}-1}}\right )}{9}\) | \(240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1666 vs.
\(2 (110) = 220\).
time = 1.21, size = 1666, normalized size = 10.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{d x^{3} \sqrt {d x^{3} - 1} - 4 \sqrt {d x^{3} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.03, size = 331, normalized size = 2.11 \begin {gather*} \frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {\left (54\,\sqrt {d\,x^3-1}+54\,\sqrt {3}-54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x\right )\,{\left (\sqrt {d\,x^3-1}-\sqrt {3}+2^{1/3}\,\sqrt {3}\,d^{1/3}\,x\right )}^3}{{\left (2^{2/3}-d^{1/3}\,x\right )}^6}\right )}{2916\,d^{2/3}}+\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {{\left (2\,\sqrt {d\,x^3-1}+2\,\sqrt {3}+2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,3{}\mathrm {i}\right )}^3\,\left (108\,\sqrt {3}-108\,\sqrt {d\,x^3-1}+54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,162{}\mathrm {i}\right )}{{\left (2^{2/3}+2\,d^{1/3}\,x-2^{2/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^6}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{2916\,d^{2/3}}+\frac {\sqrt {3}\,{314928}^{1/3}\,\ln \left (\frac {{\left (2\,\sqrt {d\,x^3-1}-2\,\sqrt {3}-2^{1/3}\,\sqrt {3}\,d^{1/3}\,x+2^{1/3}\,d^{1/3}\,x\,3{}\mathrm {i}\right )}^3\,\left (108\,\sqrt {d\,x^3-1}+108\,\sqrt {3}+54\,2^{1/3}\,\sqrt {3}\,d^{1/3}\,x-2^{1/3}\,d^{1/3}\,x\,162{}\mathrm {i}\right )}{{\left (2^{2/3}+2\,d^{1/3}\,x+2^{2/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^6}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{2916\,d^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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