Optimal. Leaf size=379 \[ -\frac {121 \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}+\frac {121 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {14 i \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {14 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {121 \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.66, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {3558, 3595, 3528, 3538, 3476, 329, 275, 200, 31, 634, 618, 204, 628, 295, 203} \[ \frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}-\frac {14 i \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {121 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}+\frac {121 \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
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 3528
Rule 3538
Rule 3558
Rule 3595
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {\int \frac {\tan ^{\frac {8}{3}}(c+d x) \left (-\frac {11 a}{3}+\frac {17}{3} i a \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \tan ^{\frac {5}{3}}(c+d x) \left (-\frac {224 i a^2}{9}-\frac {242}{9} a^2 \tan (c+d x)\right ) \, dx}{8 a^4}\\ &=-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \tan ^{\frac {2}{3}}(c+d x) \left (\frac {242 a^2}{9}-\frac {224}{9} i a^2 \tan (c+d x)\right ) \, dx}{8 a^4}\\ &=-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\frac {224 i a^2}{9}+\frac {242}{9} a^2 \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(28 i) \int \frac {1}{\sqrt [3]{\tan (c+d x)}} \, dx}{9 a^2}+\frac {121 \int \tan ^{\frac {2}{3}}(c+d x) \, dx}{36 a^2}\\ &=-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(28 i) \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 {121 \operatorname {Subst}\left (\int \frac {x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{36 a^2 d}\\ &=-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(28 i) \operatorname {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{3 a^2 d}+\frac {121 \operatorname {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{12 a^2 d}\\ &=-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(14 i) \operatorname {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d}+\frac {121 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {121 \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 {121 \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 {121 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(14 i) \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {(14 i) \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 {121 \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 {121 \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 {121 \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 {121 \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 {121 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \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 {121 \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 {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(7 i) \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 {(7 i) \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 {121 \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 {121 \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 {121 \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \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 {121 \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-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(14 i) \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 {121 \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {14 i \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {121 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \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 {121 \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-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [C] time = 3.28, size = 210, normalized size = 0.55 \[ \frac {\tan ^{\frac {2}{3}}(c+d x) \sec ^2(c+d x) \left (90 i \sqrt [3]{2} e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1}{2} \left (1-e^{2 i (c+d x)}\right )\right )+4 \left (1165 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+776 i \cos (2 (c+d x))-547 \tan (c+d x)-403 \sin (3 (c+d x)) \sec (c+d x)+344 i\right )\right )}{960 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 684, normalized size = 1.80 \[ -\frac {{\left (45 \, \sqrt {3} {\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 )} \sqrt {\frac {1}{a^{4} d^{2}}} - 45 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 45 i \, 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 (45 \, \sqrt {3} {\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 )} \sqrt {\frac {1}{a^{4} d^{2}}} + 45 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 45 i \, 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 (3495 \, \sqrt {\frac {1}{3}} {\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 )} \sqrt {\frac {1}{a^{4} d^{2}}} + 1165 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1165 i \, 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 (3495 \, \sqrt {\frac {1}{3}} {\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 )} \sqrt {\frac {1}{a^{4} d^{2}}} - 1165 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 1165 i \, 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 (2330 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 2330 i \, 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 ) - {\left (-90 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 90 i \, 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 ) - \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 (-2373 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 3837 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 555 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 45 i\right )}}{720 \, {\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 )}} \]
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 \[ \int \frac {\tan \left (d x + c\right )^{\frac {14}{3}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 390, normalized size = 1.03 \[ -\frac {3 \left (\tan ^{\frac {5}{3}}\left (d x +c \right )\right )}{5 a^{2} d}-\frac {3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{d \,a^{2}}+\frac {i}{36 d \,a^{2} \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {233 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72 d \,a^{2}}-\frac {23}{36 d \,a^{2} \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16 d \,a^{2}}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8 d \,a^{2}}+\frac {17 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{9 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}+\frac {65 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{36 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {23 \tan \left (d x +c \right )}{18 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {11 i}{18 d \,a^{2} \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {233 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144 d \,a^{2}}-\frac {233 \sqrt {3}\, \arctanh \left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72 d \,a^{2}}-\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8 d \,a^{2}} \]
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 = 5.46, size = 674, normalized size = 1.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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