Optimal. Leaf size=49 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{\sqrt {3} \sqrt {c} d} \]
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Rubi [A] time = 0.12, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2137, 203} \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{\sqrt {3} \sqrt {c} d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 2137
Rubi steps
\begin {align*} \int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx &=\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{1+3 c^3 x^2} \, dx,x,\frac {1+\frac {2 d x}{c}}{\sqrt {c^3+4 d^3 x^3}}\right )}{d}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{\sqrt {3} \sqrt {c} d}\\ \end {align*}
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Mathematica [C] time = 1.10, size = 373, normalized size = 7.61 \[ \frac {\sqrt [6]{2} \sqrt {\frac {\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (2 \sqrt {\frac {\sqrt [3]{-2} c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\sqrt [3]{-1} \left (2+\sqrt [3]{-2}\right ) c-2 \left (\sqrt [3]{-1}+2^{2/3}\right ) d x\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )-\sqrt [3]{-1} 2^{2/3} \sqrt {3} \left (1+\sqrt [3]{-1}\right ) c \sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {\frac {4 d^2 x^2}{c^2}-\frac {2 \sqrt [3]{2} d x}{c}+2^{2/3}} \Pi \left (\frac {i \sqrt [3]{2} \sqrt {3}}{2+\sqrt [3]{-2}};\sin ^{-1}\left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )\right )}{\left (2+\sqrt [3]{-2}\right ) d \sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {c^3+4 d^3 x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.04, size = 300, normalized size = 6.12 \[ \left [\frac {\sqrt {3} \sqrt {-\frac {1}{c}} \log \left (\frac {2 \, d^{6} x^{6} - 36 \, c d^{5} x^{5} - 18 \, c^{2} d^{4} x^{4} + 28 \, c^{3} d^{3} x^{3} + 18 \, c^{4} d^{2} x^{2} - c^{6} - \sqrt {3} {\left (4 \, c d^{4} x^{4} - 10 \, c^{2} d^{3} x^{3} - 18 \, c^{3} d^{2} x^{2} - 8 \, c^{4} d x - c^{5}\right )} \sqrt {4 \, d^{3} x^{3} + c^{3}} \sqrt {-\frac {1}{c}}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right )}{6 \, d}, -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (2 \, d^{3} x^{3} - 6 \, c d^{2} x^{2} - 6 \, c^{2} d x - c^{3}\right )}}{3 \, {\left (8 \, d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, c^{3} d x + c^{4}\right )} \sqrt {c}}\right )}{3 \, \sqrt {c} d}\right ] \]
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 \[ \int -\frac {2 \, d x - c}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 889, normalized size = 18.14 \[ \text {result too large to display} \]
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 \[ -\int \frac {2 \, d x - c}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 95, normalized size = 1.94 \[ \frac {\sqrt {3}\,\ln \left (\frac {{\left (-\sqrt {c^3+4\,d^3\,x^3}+\sqrt {3}\,c^{3/2}\,1{}\mathrm {i}+\sqrt {3}\,\sqrt {c}\,d\,x\,2{}\mathrm {i}\right )}^3\,\left (\sqrt {c^3+4\,d^3\,x^3}+\sqrt {3}\,c^{3/2}\,1{}\mathrm {i}+\sqrt {3}\,\sqrt {c}\,d\,x\,2{}\mathrm {i}\right )}{{\left (c+d\,x\right )}^6}\right )\,1{}\mathrm {i}}{3\,\sqrt {c}\,d} \]
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 \[ - \int \left (- \frac {c}{c \sqrt {c^{3} + 4 d^{3} x^{3}} + d x \sqrt {c^{3} + 4 d^{3} x^{3}}}\right )\, dx - \int \frac {2 d x}{c \sqrt {c^{3} + 4 d^{3} x^{3}} + d x \sqrt {c^{3} + 4 d^{3} x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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