Optimal. Leaf size=95 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{d}-\frac {\log (c+d x)}{d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2176}
\begin {gather*} -\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 )}{d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}-\frac {\log (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2176
Rubi steps
\begin {align*} \int \frac {c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx &=-\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 )}{d}-\frac {\log (c+d x)}{d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.24, size = 159, normalized size = 1.67 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2 c^3+d^3 x^3}}{4 c+2 d x+\sqrt [3]{2 c^3+d^3 x^3}}\right )}{d}+\frac {\log \left (-2 c-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{d}-\frac {\log \left (4 c^2+4 c d x+d^2 x^2+(2 c+d x) \sqrt [3]{2 c^3+d^3 x^3}+\left (2 c^3+d^3 x^3\right )^{2/3}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {-d x +c}{\left (d x +c \right ) \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {c}{c \sqrt [3]{2 c^{3} + d^{3} x^{3}} + d x \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\right )\, dx - \int \frac {d x}{c \sqrt [3]{2 c^{3} + d^{3} x^{3}} + d x \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {c-d\,x}{{\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.
[In]
[Out]
________________________________________________________________________________________