Integrand size = 26, antiderivative size = 359 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {55 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {55 i \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {8 \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {8 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {55 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {55 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {4 \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2} \]
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Time = 0.64 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {3640, 3677, 3610, 3619, 3557, 335, 215, 648, 632, 210, 642, 209, 281, 298, 31} \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {8 \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {55 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {55 i \arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}-\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {8 \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {55 i \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {55 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {4 \log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2} \]
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Rule 31
Rule 209
Rule 210
Rule 215
Rule 281
Rule 298
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3557
Rule 3610
Rule 3619
Rule 3640
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {\int \frac {\frac {14 a}{3}-\frac {8}{3} i a \tan (c+d x)}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx}{4 a^2} \\ & = \frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {\int \frac {\frac {128 a^2}{9}-\frac {110}{9} i a^2 \tan (c+d x)}{\tan ^{\frac {5}{3}}(c+d x)} \, dx}{8 a^4} \\ & = -\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {\int \frac {-\frac {110 i a^2}{9}-\frac {128}{9} a^2 \tan (c+d x)}{\tan ^{\frac {2}{3}}(c+d x)} \, dx}{8 a^4} \\ & = -\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}-\frac {(55 i) \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)} \, dx}{36 a^2}-\frac {16 \int \sqrt [3]{\tan (c+d x)} \, dx}{9 a^2} \\ & = -\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}-\frac {(55 i) \text {Subst}\left (\int \frac {1}{x^{2/3} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{36 a^2 d}-\frac {16 \text {Subst}\left (\int \frac {\sqrt [3]{x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{9 a^2 d} \\ & = -\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}-\frac {(55 i) \text {Subst}\left (\int \frac {1}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{12 a^2 d}-\frac {16 \text {Subst}\left (\int \frac {x^3}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{3 a^2 d} \\ & = -\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}-\frac {(55 i) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {(55 i) \text {Subst}\left (\int \frac {1-\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {(55 i) \text {Subst}\left (\int \frac {1+\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {8 \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d} \\ & = -\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}-\frac {(55 i) \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{144 a^2 d}-\frac {(55 i) \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{144 a^2 d}+\frac {8 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {8 \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {(55 i) \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{48 \sqrt {3} a^2 d}-\frac {(55 i) \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{48 \sqrt {3} a^2 d} \\ & = -\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {8 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {55 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {55 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {(55 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {(55 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {4 \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {4 \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d} \\ & = \frac {55 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {55 i \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {8 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {55 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {55 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {4 \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {8 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d} \\ & = \frac {55 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {55 i \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {8 \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {8 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {55 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {55 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {4 \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2} \\ \end{align*}
Time = 4.84 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {18 a^2 \sqrt [6]{\tan ^2(c+d x)}+(a+i a \tan (c+d x)) \left (66 a \sqrt [6]{\tan ^2(c+d x)}-(a+i a \tan (c+d x)) \left (-55 \left (\log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right )-\log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right )+\sqrt [3]{-1} \left (\log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )+\sqrt [3]{-1} \left (-\log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )+\log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )\right )\right )\right ) \tan (c+d x)-64 \sqrt [6]{\tan ^2(c+d x)} \left (-3+\log \left (1+\sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}+(-1)^{2/3} \log \left (1+(-1)^{2/3} \sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}\right )\right )\right )}{72 a^2 d \tan ^{\frac {2}{3}}(c+d x) \sqrt [6]{\tan ^2(c+d x)} (a+i a \tan (c+d x))^2} \]
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Time = 0.58 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}+\frac {30 i \tan \left (d x +c \right )+4 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-4 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+32}{72 {\left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}^{2}}-\frac {119 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {119 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {5 i}{12 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {1}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {119 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {3}{2 \tan \left (d x +c \right )^{\frac {2}{3}}}}{d \,a^{2}}\) | \(233\) |
default | \(\frac {\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}+\frac {30 i \tan \left (d x +c \right )+4 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-4 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+32}{72 {\left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}^{2}}-\frac {119 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {119 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {5 i}{12 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {1}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {119 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {3}{2 \tan \left (d x +c \right )^{\frac {2}{3}}}}{d \,a^{2}}\) | \(233\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 681 vs. \(2 (282) = 564\).
Time = 0.27 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {9 \, {\left (\sqrt {3} {\left (-i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\frac {1}{2} \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) + 9 \, {\left (\sqrt {3} {\left (i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (-\frac {1}{2} \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) + 119 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\frac {3}{2} \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) + 119 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (-i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (-\frac {3}{2} \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) - 238 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\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 ) - 18 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\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 ) + 3 \, \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}} {\left (103 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 75 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 31 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i\right )}}{144 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
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Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 2.00 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {119 i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{144 \, a^{2} d} - \frac {i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{16 \, a^{2} d} - \frac {\log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{16 \, a^{2} d} - \frac {119 \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{144 \, a^{2} d} + \frac {119 \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{72 \, a^{2} d} + \frac {\log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{8 \, a^{2} d} - \frac {32 \, \tan \left (d x + c\right )^{2} - 53 i \, \tan \left (d x + c\right ) - 18}{12 \, {\left (\tan \left (d x + c\right )^{\frac {4}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}^{2} a^{2} d} \]
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Time = 6.10 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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