Optimal. Leaf size=33 \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0101715, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 190, 43} \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{1+\sqrt [3]{1+x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{x}} \, dx,x,1+x\right )\\ &=3 \operatorname{Subst}\left (\int \frac{x^2}{1+x} \, dx,x,\sqrt [3]{1+x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-1+x+\frac{1}{1+x}\right ) \, dx,x,\sqrt [3]{1+x}\right )\\ &=-3 \sqrt [3]{1+x}+\frac{3}{2} (1+x)^{2/3}+3 \log \left (1+\sqrt [3]{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0120075, size = 33, normalized size = 1. \[ \frac{3}{2} (x+1)^{2/3}-3 \sqrt [3]{x+1}+3 \log \left (\sqrt [3]{x+1}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 47, normalized size = 1.4 \begin{align*} \ln \left ( 2+x \right ) +{\frac{3}{2} \left ( 1+x \right ) ^{{\frac{2}{3}}}}+2\,\ln \left ( 1+\sqrt [3]{1+x} \right ) -\ln \left ( \left ( 1+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{1+x}+1 \right ) -3\,\sqrt [3]{1+x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.955634, size = 34, normalized size = 1.03 \begin{align*} \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87864, size = 86, normalized size = 2.61 \begin{align*} \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.115146, size = 29, normalized size = 0.88 \begin{align*} \frac{3 \left (x + 1\right )^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x + 1} + 3 \log{\left (\sqrt [3]{x + 1} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.04661, size = 34, normalized size = 1.03 \begin{align*} \frac{3}{2} \,{\left (x + 1\right )}^{\frac{2}{3}} - 3 \,{\left (x + 1\right )}^{\frac{1}{3}} + 3 \, \log \left ({\left (x + 1\right )}^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]