Integrand size = 17, antiderivative size = 156 \[ \int (a+i a \tan (c+d x))^{2/3} \, dx=-\frac {a^{2/3} x}{2 \sqrt [3]{2}}+\frac {i \sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}+\frac {i a^{2/3} \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 i a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d} \]
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Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3562, 57, 631, 210, 31} \[ \int (a+i a \tan (c+d x))^{2/3} \, dx=\frac {i \sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}+\frac {3 i a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}+\frac {i a^{2/3} \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {a^{2/3} x}{2 \sqrt [3]{2}} \]
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Rule 31
Rule 57
Rule 210
Rule 631
Rule 3562
Rubi steps \begin{align*} \text {integral}& = -\frac {(i a) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {a^{2/3} x}{2 \sqrt [3]{2}}+\frac {i a^{2/3} \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {\left (3 i a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}+\frac {(3 i a) \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d} \\ & = -\frac {a^{2/3} x}{2 \sqrt [3]{2}}+\frac {i a^{2/3} \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 i a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {\left (3 i a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{\sqrt [3]{2} d} \\ & = -\frac {a^{2/3} x}{2 \sqrt [3]{2}}+\frac {i \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{2} d}+\frac {i a^{2/3} \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 i a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.71 \[ \int (a+i a \tan (c+d x))^{2/3} \, dx=\frac {i a^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\log (i+\tan (c+d x))+3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )\right )}{2 \sqrt [3]{2} d} \]
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Time = 0.55 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {3 i a \left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right )}{d}\) | \(133\) |
default | \(\frac {3 i a \left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right )}{d}\) | \(133\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (107) = 214\).
Time = 0.26 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.45 \[ \int (a+i a \tan (c+d x))^{2/3} \, dx=\frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \left (-\frac {i \, a^{2}}{2 \, d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (i \, \sqrt {3} d^{2} - d^{2}\right )} \left (-\frac {i \, a^{2}}{2 \, d^{3}}\right )^{\frac {2}{3}}}{a}\right ) + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \left (-\frac {i \, a^{2}}{2 \, d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (-i \, \sqrt {3} d^{2} - d^{2}\right )} \left (-\frac {i \, a^{2}}{2 \, d^{3}}\right )^{\frac {2}{3}}}{a}\right ) + \left (-\frac {i \, a^{2}}{2 \, d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2 \, d^{2} \left (-\frac {i \, a^{2}}{2 \, d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} a \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}}{a}\right ) \]
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\[ \int (a+i a \tan (c+d x))^{2/3} \, dx=\int \left (i a \tan {\left (c + d x \right )} + a\right )^{\frac {2}{3}}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87 \[ \int (a+i a \tan (c+d x))^{2/3} \, dx=\frac {i \, {\left (2 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {5}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 2^{\frac {2}{3}} a^{\frac {5}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 2 \cdot 2^{\frac {2}{3}} a^{\frac {5}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )\right )}}{4 \, a d} \]
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\[ \int (a+i a \tan (c+d x))^{2/3} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Time = 5.54 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.10 \[ \int (a+i a \tan (c+d x))^{2/3} \, dx=-\frac {{\left (\frac {1}{2}{}\mathrm {i}\right )}^{1/3}\,a^{2/3}\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{1/3}\right )}{d}+\frac {{\left (\frac {1}{2}{}\mathrm {i}\right )}^{1/3}\,a^{2/3}\,\ln \left (-\frac {9\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d^2}-\frac {9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{7/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,d^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d}-\frac {{\left (\frac {1}{2}{}\mathrm {i}\right )}^{1/3}\,a^{2/3}\,\ln \left (-\frac {9\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d^2}+\frac {9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{7/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,d^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d} \]
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