Optimal. Leaf size=204 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \]
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Rubi [A]
time = 0.03, antiderivative size = 204, 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 = {401}
\begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c+3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 401
Rubi steps
\begin {align*} \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 6.15, size = 153, normalized size = 0.75 \begin {gather*} \frac {3 c x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2} \left (3 c F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )+2 d x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-d \,x^{2}+c \right ) \left (3 d \,x^{2}+c \right )^{\frac {1}{3}}}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{- c \sqrt [3]{c + 3 d x^{2}} + d x^{2} \sqrt [3]{c + 3 d x^{2}}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (c-d\,x^2\right )\,{\left (3\,d\,x^2+c\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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