Integrand size = 26, antiderivative size = 319 \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {5 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {5 \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac {2 i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{\sqrt {3} a d}+\frac {5 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {5 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {5 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{3 a d}-\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))} \]
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Time = 0.57 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3631, 3609, 3619, 3557, 335, 281, 206, 31, 648, 632, 210, 642, 301, 209} \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {2 i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{\sqrt {3} a d}-\frac {5 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {5 \arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{12 a d}+\frac {5 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}+\frac {2 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{3 a d}+\frac {5 \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt {3} a d}-\frac {5 \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt {3} a d}-\frac {i \log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{3 a d} \]
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Rule 31
Rule 206
Rule 209
Rule 210
Rule 281
Rule 301
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3557
Rule 3609
Rule 3619
Rule 3631
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \tan ^{\frac {2}{3}}(c+d x) \left (\frac {5 a}{3}-\frac {8}{3} i a \tan (c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {\int \frac {\frac {8 i a}{3}+\frac {5}{3} a \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {(4 i) \int \frac {1}{\sqrt [3]{\tan (c+d x)}} \, dx}{3 a}+\frac {5 \int \tan ^{\frac {2}{3}}(c+d x) \, dx}{6 a} \\ & = -\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {(4 i) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 a d}+\frac {5 \text {Subst}\left (\int \frac {x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{6 a d} \\ & = -\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {(4 i) \text {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 a d} \\ & = -\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{a d}+\frac {5 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {5 \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {5 \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d} \\ & = \frac {5 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {(2 i) \text {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {5 \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{24 a d}+\frac {5 \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{24 a d}+\frac {5 \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{8 \sqrt {3} a d}-\frac {5 \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{8 \sqrt {3} a d} \\ & = \frac {5 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {5 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {5 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {i \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {i \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{a d}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d} \\ & = -\frac {5 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {5 \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {5 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {5 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {5 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{3 a d}-\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {(2 i) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac {2}{3}}(c+d x)\right )}{a d} \\ & = -\frac {5 \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {5 \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac {2 i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{\sqrt {3} a d}+\frac {5 \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {5 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {5 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{3 a d}-\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{a d}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))} \\ \end{align*}
Time = 5.79 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.06 \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\frac {6 \tan ^{\frac {11}{3}}(c+d x)}{a+i a \tan (c+d x)}+\frac {2 i \left (4 \sqrt {3} \arctan \left (\frac {-1+2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )+4 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )-2 \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )-12 \tan ^{\frac {2}{3}}(c+d x)+3 \tan ^{\frac {8}{3}}(c+d x)\right )}{a}+\frac {\tan ^{\frac {5}{3}}(c+d x) \left (5 i \log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right )-5 i \log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right )+5 \sqrt [6]{-1} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )-5 \sqrt [6]{-1} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )+5 (-1)^{5/6} \log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )-5 (-1)^{5/6} \log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )-6 \tan ^2(c+d x)^{5/6}\right )}{a \tan ^2(c+d x)^{5/6}}}{12 d} \]
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Time = 0.51 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{2}+\frac {13 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{12}-\frac {1}{6 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{4}-\frac {4 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )-2 i}{12 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}-\frac {13 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{24}-\frac {13 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{12}-\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{8}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{4}}{d a}\) | \(201\) |
default | \(\frac {-\frac {3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{2}+\frac {13 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{12}-\frac {1}{6 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{4}-\frac {4 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )-2 i}{12 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}-\frac {13 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{24}-\frac {13 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{12}-\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{8}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{4}}{d a}\) | \(201\) |
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Time = 0.26 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.55 \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {{\left (3 \, {\left (\sqrt {3} a d \sqrt {\frac {1}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (\frac {1}{2} \, \sqrt {3} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - 3 \, {\left (\sqrt {3} a d \sqrt {\frac {1}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (-\frac {1}{2} \, \sqrt {3} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - 13 \, {\left (3 \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {1}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (\frac {3}{2} \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + 13 \, {\left (3 \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {1}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (-\frac {3}{2} \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + 26 i \, e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + i\right ) + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - i\right ) - 6 \, \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{24 \, a d} \]
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\[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\tan ^{\frac {8}{3}}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
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Exception generated. \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.58 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.72 \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {13 \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{24 \, a d} - \frac {\sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{8 \, a d} - \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{8 \, a d} - \frac {13 i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{24 \, a d} + \frac {13 i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{12 \, a d} + \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{4 \, a d} - \frac {3 i \, \tan \left (d x + c\right )^{\frac {2}{3}}}{2 \, a d} - \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{2 \, a d {\left (\tan \left (d x + c\right ) - i\right )}} \]
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Time = 7.32 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.93 \[ \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx=\ln \left (\left (a^3\,d^3\,312480{}\mathrm {i}-a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{1/3}\,307584{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{2/3}+24336\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{1/3}+\ln \left (\left (a^3\,d^3\,312480{}\mathrm {i}-a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{1/3}\,307584{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{2/3}+24336\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{1/3}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}\,3{}\mathrm {i}}{2\,a\,d}+\frac {\ln \left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,312480{}\mathrm {i}-a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{1/3}\,153792{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{2/3}}{4}+24336\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,312480{}\mathrm {i}+a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{1/3}\,153792{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{2/3}}{4}+24336\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{64\,a^3\,d^3}\right )}^{1/3}}{2}+\frac {\ln \left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,312480{}\mathrm {i}-a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{1/3}\,153792{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{2/3}}{4}+24336\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,312480{}\mathrm {i}+a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{1/3}\,153792{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{2/3}}{4}+24336\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {2197{}\mathrm {i}}{1728\,a^3\,d^3}\right )}^{1/3}}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}\,1{}\mathrm {i}}{2\,a\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \]
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