Optimal. Leaf size=323 \[ -\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (c-\left (1+\sqrt{3}\right ) d\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\left (1-\sqrt{3}\right ) d\right )} \]
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Rubi [A] time = 0.854269, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2142, 2113, 537, 571, 93, 205} \[ -\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (c-\left (1+\sqrt{3}\right ) d\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\left (1-\sqrt{3}\right ) d\right )} \]
Antiderivative was successfully verified.
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Rule 2142
Rule 2113
Rule 537
Rule 571
Rule 93
Rule 205
Rubi steps
\begin{align*} \int \frac{1+\sqrt{3}+x}{(c+d x) \sqrt{-1-x^3}} \, dx &=\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-c+\left (1-\sqrt{3}\right ) d+\left (-c+\left (1+\sqrt{3}\right ) d\right ) x\right ) \sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx,x,\frac{-1+\sqrt{3}-x}{1+\sqrt{3}+x}\right )}{\sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ &=-\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (-c+d+\sqrt{3} d\right ) (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2} \left (\left (-c+\left (1-\sqrt{3}\right ) d\right )^2-\left (-c+\left (1+\sqrt{3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac{-1+\sqrt{3}-x}{1+\sqrt{3}+x}\right )}{\sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}+\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (-c+\left (1-\sqrt{3}\right ) d\right ) (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2} \left (\left (-c+\left (1-\sqrt{3}\right ) d\right )^2-\left (-c+\left (1+\sqrt{3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac{-1+\sqrt{3}-x}{1+\sqrt{3}+x}\right )}{\sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ &=-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{\left (c-\left (1-\sqrt{3}\right ) d\right ) \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}-\frac{\left (2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (-c+d+\sqrt{3} d\right ) (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{7-4 \sqrt{3}+x} \left (\left (-c+\left (1-\sqrt{3}\right ) d\right )^2-\left (-c+\left (1+\sqrt{3}\right ) d\right )^2 x\right )} \, dx,x,\frac{\left (-1+\sqrt{3}-x\right )^2}{\left (1+\sqrt{3}+x\right )^2}\right )}{\sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ &=-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{\left (c-\left (1-\sqrt{3}\right ) d\right ) \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}-\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (-c+d+\sqrt{3} d\right ) (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\left (-c+\left (1-\sqrt{3}\right ) d\right )^2+\left (-c+\left (1+\sqrt{3}\right ) d\right )^2-\left (\left (-c+\left (1-\sqrt{3}\right ) d\right )^2+\left (7-4 \sqrt{3}\right ) \left (-c+\left (1+\sqrt{3}\right ) d\right )^2\right ) x^2} \, dx,x,\frac{\sqrt [4]{3} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}}}{\sqrt{-\frac{\left (-2+\sqrt{3}\right ) \left (1-x+x^2\right )}{\left (1+\sqrt{3}+x\right )^2}}}\right )}{\sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ &=-\frac{\left (c-d-\sqrt{3} d\right ) (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} \tan ^{-1}\left (\frac{\sqrt{c^2+c d+d^2} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}}}{\sqrt{c-d} \sqrt{d} \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}}}\right )}{\sqrt{c-d} \sqrt{d} \sqrt{c^2+c d+d^2} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{\left (c-\left (1-\sqrt{3}\right ) d\right ) \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ \end{align*}
Mathematica [C] time = 0.251142, size = 216, normalized size = 0.67 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left (c-\left (1+\sqrt{3}\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt{-x^3-1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.039, size = 266, normalized size = 0.8 \begin{align*}{\frac{-{\frac{2\,i}{3}}\sqrt{3}}{d}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{{\frac{2\,i}{3}} \left ( d\sqrt{3}-c+d \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + \sqrt{3} + 1}{\sqrt{-x^{3} - 1}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1 + \sqrt{3}}{\sqrt{- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + \sqrt{3} + 1}{\sqrt{-x^{3} - 1}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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