Optimal. Leaf size=339 \[ \frac {7 i \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {7 i \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}-\frac {7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}-\frac {2 \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {2 \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}-\frac {7 i \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}+\frac {7 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {\log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.59, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3559, 3596, 3538, 3476, 329, 275, 200, 31, 634, 618, 204, 628, 295, 203} \[ \frac {7 i \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {7 i \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}-\frac {2 \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {2 \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}-\frac {7 i \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}+\frac {7 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {\log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 203
Rule 204
Rule 275
Rule 295
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3476
Rule 3538
Rule 3559
Rule 3596
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx &=\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\frac {10 a}{3}-\frac {4}{3} i a \tan (c+d x)}{\sqrt [3]{\tan (c+d x)} (a+i a \tan (c+d x))} \, dx}{4 a^2}\\ &=\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\frac {32 a^2}{9}-\frac {14}{9} i a^2 \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{8 a^4}\\ &=\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \int \tan ^{\frac {2}{3}}(c+d x) \, dx}{36 a^2}+\frac {4 \int \frac {1}{\sqrt [3]{\tan (c+d x)}} \, dx}{9 a^2}\\ &=\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \operatorname {Subst}\left (\int \frac {x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{36 a^2 d}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{9 a^2 d}\\ &=\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \operatorname {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{12 a^2 d}+\frac {4 \operatorname {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{3 a^2 d}\\ &=\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {(7 i) \operatorname {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 )}{36 a^2 d}-\frac {(7 i) \operatorname {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 )}{36 a^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d}\\ &=-\frac {7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \operatorname {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 {(7 i) \operatorname {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 {2 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {(7 i) \operatorname {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 {(7 i) \operatorname {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 {7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {2 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {7 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 {7 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 {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(7 i) \operatorname {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 {(7 i) \operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d}\\ &=\frac {7 i \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {7 i \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {2 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {7 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 {7 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 {\log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}+\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {2 \operatorname {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 {7 i \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {7 i \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {2 \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {2 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {7 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 {7 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 {\log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}+\frac {7 \tan ^{\frac {2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}+\frac {\tan ^{\frac {2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [C] time = 1.53, size = 189, normalized size = 0.56 \[ -\frac {\tan ^{\frac {2}{3}}(c+d x) \sec ^2(c+d x) \left (9 \sqrt [3]{2} e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1}{2} \left (1-e^{2 i (c+d x)}\right )\right )+46 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+28 i \sin (2 (c+d x))+40 \cos (2 (c+d x))+40\right )}{96 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 519, normalized size = 1.53 \[ \frac {{\left ({\left (-9 i \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} - 9 \, 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 ) + {\left (9 i \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} - 9 \, 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 ) + {\left (69 i \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} - 23 \, 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 ) + {\left (-69 i \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} - 23 \, 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 ) + 46 \, e^{\left (4 i \, d x + 4 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 ) + 18 \, e^{\left (4 i \, d x + 4 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 ) + 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 {2}{3}} {\left (17 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 20 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{144 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 355, normalized size = 1.05 \[ -\frac {7 i}{36 d \,a^{2} \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {1}{36 d \,a^{2} \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {23 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72 d \,a^{2}}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16 d \,a^{2}}-\frac {i \sqrt {3}\, \arctanh \left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8 d \,a^{2}}-\frac {11 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{18 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}+\frac {25 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{36 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {7 i \tan \left (d x +c \right )}{18 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}+\frac {2}{9 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {23 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144 d \,a^{2}}+\frac {23 i \sqrt {3}\, \arctanh \left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72 d \,a^{2}}+\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 630, normalized size = 1.86 \[ \frac {23\,\ln \left (\frac {529\,\left (\frac {1795840\,a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {1}{a^6\,d^3}\right )}^{1/3}}{3}+\frac {a^6\,d^3\,1464064{}\mathrm {i}}{3}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{2/3}}{5184}-\frac {33856\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{1/3}}{72}+\ln \left (\left (1873920\,a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{1/3}+\frac {a^6\,d^3\,1464064{}\mathrm {i}}{3}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{2/3}-\frac {33856\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{1/3}-\frac {-\frac {7\,{\mathrm {tan}\left (c+d\,x\right )}^{5/3}}{12\,a^2\,d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}\,5{}\mathrm {i}}{6\,a^2\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}}+\frac {23\,\ln \left (-\frac {33856\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+\frac {529\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {a^6\,d^3\,1464064{}\mathrm {i}}{3}+\frac {897920\,a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{1/3}}{3}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{2/3}}{20736}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{1/3}}{144}-\frac {23\,\ln \left (-\frac {33856\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+\frac {529\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {a^6\,d^3\,1464064{}\mathrm {i}}{3}-\frac {897920\,a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{1/3}}{3}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{2/3}}{20736}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^6\,d^3}\right )}^{1/3}}{144}+\ln \left (-\frac {33856\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left (\frac {a^6\,d^3\,1464064{}\mathrm {i}}{3}+1873920\,a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{1/3}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{2/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{1/3}-\ln \left (-\frac {33856\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left (\frac {a^6\,d^3\,1464064{}\mathrm {i}}{3}-1873920\,a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{1/3}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{2/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{512\,a^6\,d^3}\right )}^{1/3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{\tan ^{\frac {7}{3}}{\left (c + d x \right )} - 2 i \tan ^{\frac {4}{3}}{\left (c + d x \right )} - \sqrt [3]{\tan {\left (c + d x \right )}}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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