Optimal. Leaf size=379 \[ -\frac {121 \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \text {ArcTan}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {14 i \text {ArcTan}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {121 \text {ArcTan}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {14 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {121 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {121 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {7 i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {14 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d}-\frac {121 \tan ^{\frac {5}{3}}(c+d x)}{60 a^2 d}+\frac {7 i \tan ^{\frac {8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A]
time = 0.46, 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 = {3639, 3676, 3609, 3619, 3557, 335, 281, 206, 31, 648, 632, 210, 642, 301, 209}
\begin {gather*} -\frac {14 i \text {ArcTan}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {121 \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {121 \text {ArcTan}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}+\frac {121 \text {ArcTan}\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 \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} \end {gather*}
Antiderivative was successfully verified.
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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 3639
Rule 3676
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) \text {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 \text {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) \text {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{3 a^2 d}+\frac {121 \text {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) \text {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a^2 d}+\frac {121 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {121 \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 )}{36 a^2 d}+\frac {121 \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 )}{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) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {(14 i) \text {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 \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 {121 \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 {121 \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 {121 \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 {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) \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 {(7 i) \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 {121 \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 {121 \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 {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) \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 {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 3.59, size = 210, normalized size = 0.55 \begin {gather*} \frac {\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 (344 i+776 i \cos (2 (c+d x))+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 ) (-i \cos (2 (c+d x))+\sin (2 (c+d x)))-403 \sec (c+d x) \sin (3 (c+d x))-547 \tan (c+d x)\right )\right ) \tan ^{\frac {2}{3}}(c+d x)}{960 a^2 d (-i+\tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 246, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {-\frac {3 \left (\tan ^{\frac {5}{3}}\left (d x +c \right )\right )}{5}-3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )+\frac {i}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {233 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {23}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {92 \tan \left (d x +c \right )-136 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-130 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+44 i}{72 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {233 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {233 \sqrt {3}\, \arctanh \left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}-\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}}{d \,a^{2}}\) | \(246\) |
default | \(\frac {-\frac {3 \left (\tan ^{\frac {5}{3}}\left (d x +c \right )\right )}{5}-3 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )+\frac {i}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {233 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {23}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {92 \tan \left (d x +c \right )-136 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-130 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+44 i}{72 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}-\frac {233 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {233 \sqrt {3}\, \arctanh \left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}-\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}}{d \,a^{2}}\) | \(246\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 684 vs. \(2 (298) = 596\).
time = 0.63, size = 684, normalized size = 1.80 \begin {gather*} -\frac {45 \, {\left (\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}}} - i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 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 ) - 45 \, {\left (\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}}} + i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 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 ) + 1165 \, {\left (3 \, \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}}} + i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 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 ) - 1165 \, {\left (3 \, \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}}} - i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 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 ) + 2330 \, {\left (-i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 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 ) + 90 \, {\left (i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 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 ) + 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 (791 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1279 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 185 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 15 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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 268, normalized size = 0.71 \begin {gather*} \frac {233 \, \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 {\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 {i \, \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 {233 i \, \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 {233 i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{72 \, a^{2} d} - \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{8 \, a^{2} d} - \frac {23 \, \tan \left (d x + c\right )^{\frac {5}{3}} - 20 i \, \tan \left (d x + c\right )^{\frac {2}{3}}}{12 \, a^{2} d {\left (\tan \left (d x + c\right ) - i\right )}^{2}} - \frac {3 \, {\left (a^{8} d^{4} \tan \left (d x + c\right )^{\frac {5}{3}} + 5 i \, a^{8} d^{4} \tan \left (d x + c\right )^{\frac {2}{3}}\right )}}{5 \, a^{10} d^{5}} \end {gather*}
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 \begin {gather*} \ln \left (\left (\frac {a^6\,d^3\,1619208448{}\mathrm {i}}{3}-a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{1/3}\,167024640{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{2/3}+\frac {24321472\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{1/3}+\ln \left (\left (\frac {a^6\,d^3\,1619208448{}\mathrm {i}}{3}-a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}\,167024640{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{2/3}+\frac {24321472\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}-\frac {\frac {5\,{\mathrm {tan}\left (c+d\,x\right )}^{2/3}}{3\,a^2\,d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^{5/3}\,23{}\mathrm {i}}{12\,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 {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}\,3{}\mathrm {i}}{a^2\,d}-\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^{5/3}}{5\,a^2\,d}+\frac {\ln \left (\frac {24321472\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {a^6\,d^3\,1619208448{}\mathrm {i}}{3}-a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{1/3}\,83512320{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{2/3}}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (\frac {24321472\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {a^6\,d^3\,1619208448{}\mathrm {i}}{3}+a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{1/3}\,83512320{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{2/3}}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{1/3}}{2}+\frac {\ln \left (\frac {24321472\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {a^6\,d^3\,1619208448{}\mathrm {i}}{3}-a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}\,83512320{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{2/3}}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (\frac {24321472\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {a^6\,d^3\,1619208448{}\mathrm {i}}{3}+a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}\,83512320{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{2/3}}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {12649337{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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