Optimal. Leaf size=186 \[ -\frac {\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac {\tan ^{-1}\left (\frac {\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} c d}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 c d}-\frac {\log (c+d x)}{2 c d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, 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 = {2149, 239, 2151} \[ -\frac {\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac {\tan ^{-1}\left (\frac {\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} c d}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 c d}-\frac {\log (c+d x)}{2 c d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 239
Rule 2149
Rule 2151
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*}
________________________________________________________________________________________
Mathematica [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (2\,c^3+d^3\,x^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c + d x\right ) \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________