Optimal. Leaf size=186 \[ \frac {\tan ^{-1}\left (\frac {1+\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3} c d}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 c d}-\frac {\log (c+d x)}{2 c d}-\frac {\log \left (-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d} \]
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Rubi [A]
time = 0.13, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2175, 245,
2176} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} c d}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 c d}-\frac {\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}-\frac {\log (c+d x)}{2 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 2175
Rule 2176
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx &=\frac {\int \frac {1}{\sqrt [3]{2 c^3+d^3 x^3}} \, dx}{2 c}+\frac {\int \frac {c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx}{2 c}\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3} c d}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 c d}-\frac {\log (c+d x)}{2 c d}-\frac {\log \left (-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}\\ \end {align*}
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Mathematica [F]
time = 7.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {1}{3}}}\, dx\]
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}{\left (c + d x\right ) \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{{\left (2\,c^3+d^3\,x^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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