Optimal. Leaf size=208 \[ \frac{\log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\sqrt [3]{a}\right )}{3 a^{2/3} \sqrt [3]{b c-a d}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b c-a d}}-\frac{\log \left (a^{2/3}-\frac{\sqrt [3]{a} x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left (c+d x^3\right )^{2/3}}\right )}{6 a^{2/3} \sqrt [3]{b c-a d}} \]
[Out]
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Rubi [A] time = 0.457942, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\sqrt [3]{a}\right )}{3 a^{2/3} \sqrt [3]{b c-a d}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b c-a d}}-\frac{\log \left (a^{2/3}-\frac{\sqrt [3]{a} x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left (c+d x^3\right )^{2/3}}\right )}{6 a^{2/3} \sqrt [3]{b c-a d}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^3)*(c + d*x^3)^(1/3)),x]
[Out]
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Rubi in Sympy [A] time = 34.6177, size = 185, normalized size = 0.89 \[ - \frac{\log{\left (\sqrt [3]{a} - \frac{x \sqrt [3]{a d - b c}}{\sqrt [3]{c + d x^{3}}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{a d - b c}} + \frac{\log{\left (a^{\frac{2}{3}} + \frac{\sqrt [3]{a} x \sqrt [3]{a d - b c}}{\sqrt [3]{c + d x^{3}}} + \frac{x^{2} \left (a d - b c\right )^{\frac{2}{3}}}{\left (c + d x^{3}\right )^{\frac{2}{3}}} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{a d - b c}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{a d - b c}}{3 \sqrt [3]{c + d x^{3}}}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**3+a)/(d*x**3+c)**(1/3),x)
[Out]
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Mathematica [A] time = 0.344175, size = 168, normalized size = 0.81 \[ \frac{\log \left (a^{2/3}+\frac{\sqrt [3]{a} x \sqrt [3]{a d-b c}}{\sqrt [3]{c x^3+d}}+\frac{x^2 (a d-b c)^{2/3}}{\left (c x^3+d\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{a}-\frac{x \sqrt [3]{a d-b c}}{\sqrt [3]{c x^3+d}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a d-b c}}{\sqrt [3]{a} \sqrt [3]{c x^3+d}}+1}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{a d-b c}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^3)*(c + d*x^3)^(1/3)),x]
[Out]
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Maple [F] time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{3}+a}{\frac{1}{\sqrt [3]{d{x}^{3}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^3+a)/(d*x^3+c)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)^(1/3)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)^(1/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{3}\right ) \sqrt [3]{c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**3+a)/(d*x**3+c)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)^(1/3)),x, algorithm="giac")
[Out]