Integrand size = 28, antiderivative size = 202 \[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=-\frac {a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}+\frac {\sqrt {3} a^{2/3} (i A+B) \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}+\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}+\frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d} \]
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Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3608, 3562, 57, 631, 210, 31} \[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {3} a^{2/3} (B+i A) \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}+\frac {3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}+\frac {a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}+\frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d} \]
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Rule 31
Rule 57
Rule 210
Rule 631
Rule 3562
Rule 3608
Rubi steps \begin{align*} \text {integral}& = \frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d}-(-A+i B) \int (a+i a \tan (c+d x))^{2/3} \, dx \\ & = \frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d}-\frac {(a (i A+B)) \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} (A-i B) x}{2 \sqrt [3]{2}}+\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d}-\frac {\left (3 a^{2/3} (i A+B)\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 a (i A+B)) \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} (A-i B) x}{2 \sqrt [3]{2}}+\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}+\frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d}-\frac {\left (3 a^{2/3} (i A+B)\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} (A-i B) x}{2 \sqrt [3]{2}}+\frac {\sqrt {3} a^{2/3} (i A+B) \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 {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}+\frac {3 B (a+i a \tan (c+d x))^{2/3}}{2 d} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.68 \[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {2^{2/3} a^{2/3} (i A+B) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\log (i+\tan (c+d x))+3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )\right )+6 B (a+i a \tan (c+d x))^{2/3}}{4 d} \]
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Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {3 i \left (-\frac {i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2}+\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) a \left (-i B +A \right )\right )}{d}\) | \(159\) |
default | \(\frac {3 i \left (-\frac {i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2}+\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) a \left (-i B +A \right )\right )}{d}\) | \(159\) |
parts | \(\frac {3 i A a \left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right )}{d}+B \left (\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2 d}+\frac {a^{\frac {2}{3}} 2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{2 d}-\frac {a^{\frac {2}{3}} 2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{4 d}+\frac {a^{\frac {2}{3}} \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{2 d}\right )\) | \(290\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (146) = 292\).
Time = 0.27 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.42 \[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {3 \cdot 2^{\frac {2}{3}} B \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} e^{\left (\frac {4}{3} i \, d x + \frac {4}{3} i \, c\right )} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} d \left (\frac {{\left (-i \, A^{3} - 3 \, A^{2} B + 3 i \, A B^{2} + B^{3}\right )} a^{2}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} {\left (A^{2} - 2 i \, A B - B^{2}\right )} 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 )} + 2 \, \left (\frac {1}{2}\right )^{\frac {2}{3}} d^{2} \left (\frac {{\left (-i \, A^{3} - 3 \, A^{2} B + 3 i \, A B^{2} + B^{3}\right )} a^{2}}{d^{3}}\right )^{\frac {2}{3}}}{{\left (A^{2} - 2 i \, A B - B^{2}\right )} a}\right ) + \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} d - d\right )} \left (\frac {{\left (-i \, A^{3} - 3 \, A^{2} B + 3 i \, A B^{2} + B^{3}\right )} a^{2}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} {\left (A^{2} - 2 i \, A B - B^{2}\right )} 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 (\frac {1}{2}\right )^{\frac {2}{3}} {\left (i \, \sqrt {3} d^{2} + d^{2}\right )} \left (\frac {{\left (-i \, A^{3} - 3 \, A^{2} B + 3 i \, A B^{2} + B^{3}\right )} a^{2}}{d^{3}}\right )^{\frac {2}{3}}}{{\left (A^{2} - 2 i \, A B - B^{2}\right )} a}\right ) + \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} d - d\right )} \left (\frac {{\left (-i \, A^{3} - 3 \, A^{2} B + 3 i \, A B^{2} + B^{3}\right )} a^{2}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} {\left (A^{2} - 2 i \, A B - B^{2}\right )} 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 (\frac {1}{2}\right )^{\frac {2}{3}} {\left (-i \, \sqrt {3} d^{2} + d^{2}\right )} \left (\frac {{\left (-i \, A^{3} - 3 \, A^{2} B + 3 i \, A B^{2} + B^{3}\right )} a^{2}}{d^{3}}\right )^{\frac {2}{3}}}{{\left (A^{2} - 2 i \, A B - B^{2}\right )} a}\right )}{2 \, d} \]
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\[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {2}{3}} \left (A + B \tan {\left (c + d x \right )}\right )\, dx \]
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Time = 0.38 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83 \[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {i \, {\left (2 \, \sqrt {3} 2^{\frac {2}{3}} {\left (A - i \, B\right )} a^{\frac {5}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 2^{\frac {2}{3}} {\left (A - i \, B\right )} a^{\frac {5}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 2 \cdot 2^{\frac {2}{3}} {\left (A - i \, B\right )} a^{\frac {5}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 6 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} B a\right )}}{4 \, a d} \]
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\[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Time = 9.38 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.82 \[ \int (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {3\,B\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{2/3}}{2\,d}+\frac {2^{2/3}\,B\,a^{2/3}\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,a^{1/3}\right )}{2\,d}-\frac {{\left (\frac {1}{2}{}\mathrm {i}\right )}^{1/3}\,A\,a^{2/3}\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{1/3}\right )}{d}-\frac {{\left (\frac {1}{2}{}\mathrm {i}\right )}^{1/3}\,A\,a^{2/3}\,\ln \left (\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{1/3}}{2}-{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {{\left (-1\right )}^{5/6}\,2^{1/3}\,\sqrt {3}\,a^{1/3}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d}+\frac {2^{2/3}\,B\,a^{2/3}\,\ln \left (\frac {9\,B^2\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d^2}-\frac {9\,2^{1/3}\,B^2\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{d^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {2^{2/3}\,B\,a^{2/3}\,\ln \left (\frac {9\,B^2\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d^2}-\frac {9\,2^{1/3}\,B^2\,a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{d^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (\frac {1}{2}{}\mathrm {i}\right )}^{1/3}\,A\,a^{2/3}\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{1/3}}{2}+\frac {{\left (-1\right )}^{5/6}\,2^{1/3}\,\sqrt {3}\,a^{1/3}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d} \]
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