Optimal. Leaf size=337 \[ \frac {25 \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {25 \text {ArcTan}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {2 i \text {ArcTan}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {25 \text {ArcTan}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {25 \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}+\frac {25 \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}+\frac {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}+\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A]
time = 0.40, antiderivative size = 337, 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 = {3639, 3676,
3619, 3557, 335, 281, 206, 31, 648, 632, 210, 642, 301, 209} \begin {gather*} \frac {2 i \text {ArcTan}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}+\frac {25 \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {25 \text {ArcTan}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{72 a^2 d}-\frac {25 \text {ArcTan}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {2 i \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{9 a^2 d}-\frac {25 \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 {25 \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 {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 {5}{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 3619
Rule 3639
Rule 3676
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {8}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {\int \frac {\tan ^{\frac {2}{3}}(c+d x) \left (-\frac {5 a}{3}+\frac {11}{3} i a \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {-\frac {32 i a^2}{9}-\frac {50}{9} a^2 \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{8 a^4}\\ &=\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(4 i) \int \frac {1}{\sqrt [3]{\tan (c+d x)}} \, dx}{9 a^2}-\frac {25 \int \tan ^{\frac {2}{3}}(c+d x) \, dx}{36 a^2}\\ &=\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(4 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 {25 \text {Subst}\left (\int \frac {x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{36 a^2 d}\\ &=\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(4 i) \text {Subst}\left (\int \frac {x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{3 a^2 d}-\frac {25 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{12 a^2 d}\\ &=\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(2 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 {25 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {25 \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 {25 \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 {25 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {(2 i) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 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 )}{9 a^2 d}-\frac {25 \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 {25 \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 {25 \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 {25 \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 {25 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {25 \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 {25 \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 {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {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 {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 {25 \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 {25 \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 {25 \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {25 \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {25 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {25 \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 {25 \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 {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}+\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(2 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 {25 \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {25 \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {2 i \tan ^{-1}\left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {25 \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}-\frac {2 i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}-\frac {25 \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 {25 \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 {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}+\frac {2 i \tan ^{\frac {2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}-\frac {\tan ^{\frac {5}{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 = 1.76, size = 194, normalized size = 0.58 \begin {gather*} -\frac {i \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 )-82 \, _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)))+4 (8+8 \cos (2 (c+d x))+11 i \sin (2 (c+d x)))\right ) \tan ^{\frac {2}{3}}(c+d x)}{96 a^2 d (-i+\tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 225, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {-\frac {41 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {i}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {11}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}-\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}+\frac {\sqrt {3}\, \arctanh \left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}+\frac {44 \tan \left (d x +c \right )-64 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-58 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+20 i}{72 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}+\frac {41 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}+\frac {41 \sqrt {3}\, \arctanh \left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}+\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}}{d \,a^{2}}\) | \(225\) |
default | \(\frac {-\frac {41 i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {i}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {11}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}-\frac {i \ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}+\frac {\sqrt {3}\, \arctanh \left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}+\frac {44 \tan \left (d x +c \right )-64 i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-58 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+20 i}{72 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )^{2}}+\frac {41 i \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}+\frac {41 \sqrt {3}\, \arctanh \left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}+\frac {i \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}}{d \,a^{2}}\) | \(225\) |
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 [A]
time = 0.72, size = 521, normalized size = 1.55 \begin {gather*} \frac {{\left (9 \, {\left (\sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 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 ) - 9 \, {\left (\sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 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 ) + 41 \, {\left (3 \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 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 ) - 41 \, {\left (3 \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 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 ) - 82 i \, 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 i \, 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 (-19 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 16 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{144 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 228, normalized size = 0.68 \begin {gather*} -\frac {41 \, \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 {41 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 {41 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 {11 \, \tan \left (d x + c\right )^{\frac {5}{3}} - 8 i \, \tan \left (d x + c\right )^{\frac {2}{3}}}{12 \, a^{2} d {\left (\tan \left (d x + c\right ) - i\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.26, size = 642, normalized size = 1.91 \begin {gather*} \ln \left (-\left (\frac {a^6\,d^3\,8915200{}\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}\,5412864{}\mathrm {i}\right )\,{\left (-\frac {1{}\mathrm {i}}{512\,a^6\,d^3}\right )}^{2/3}-\frac {107584\,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\,8915200{}\mathrm {i}}{3}+a^8\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}\,5412864{}\mathrm {i}\right )\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{2/3}-\frac {107584\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{3}\right )\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}+\frac {\frac {2\,{\mathrm {tan}\left (c+d\,x\right )}^{2/3}}{3\,a^2\,d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^{5/3}\,11{}\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 {\ln \left (-\frac {107584\,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\,8915200{}\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}\,2706432{}\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 {107584\,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\,8915200{}\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}\,2706432{}\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 {107584\,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\,8915200{}\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 {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}\,2706432{}\mathrm {i}\right )\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{2/3}}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}}{2}-\frac {\ln \left (-\frac {107584\,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\,8915200{}\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 {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}\,2706432{}\mathrm {i}\right )\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{2/3}}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {68921{}\mathrm {i}}{373248\,a^6\,d^3}\right )}^{1/3}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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