Optimal. Leaf size=237 \[ -\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d} \]
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Rubi [A]
time = 0.20, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3641, 3673,
3607, 3562, 57, 631, 210, 31} \begin {gather*} -\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}+\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 210
Rule 631
Rule 3562
Rule 3607
Rule 3641
Rule 3673
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx &=\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {3 \int \frac {\tan (c+d x) \left (2 a-\frac {1}{3} i a \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{5 a}\\ &=\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}-\frac {3 \int \frac {\frac {i a}{3}+2 a \tan (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{5 a}\\ &=\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}+\frac {i \int (a+i a \tan (c+d x))^{2/3} \, dx}{2 a}\\ &=\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}+\frac {\text {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{2 d}\\ &=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}-\frac {3 \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 d}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {21}{10 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 \tan ^2(c+d x)}{5 d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {3 (a+i a \tan (c+d x))^{2/3}}{10 a d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.90, size = 115, normalized size = 0.49 \begin {gather*} \frac {3 \sec ^2(c+d x) \left (40+24 \cos (2 (c+d x))+5 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) (1+\cos (2 (c+d x))+i \sin (2 (c+d x)))+4 i \sin (2 (c+d x))\right )}{80 d \sqrt [3]{a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 192, normalized size = 0.81
method | result | size |
derivativedivides | \(-\frac {3 \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{3}}}{5}-\frac {a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2}+\frac {\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) a^{2}}{2}-\frac {a^{2}}{2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}\right )}{d \,a^{2}}\) | \(192\) |
default | \(-\frac {3 \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{3}}}{5}-\frac {a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2}+\frac {\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) a^{2}}{2}-\frac {a^{2}}{2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}\right )}{d \,a^{2}}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 190, normalized size = 0.80 \begin {gather*} -\frac {10 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {11}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 5 \cdot 2^{\frac {2}{3}} a^{\frac {11}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 10 \cdot 2^{\frac {2}{3}} a^{\frac {11}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) + 24 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}} a^{2} - 60 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} a^{3} - \frac {60 \, a^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}}{40 \, a^{4} d} \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. 418 vs. \(2 (172) = 344\).
time = 1.49, size = 418, normalized size = 1.76 \begin {gather*} \frac {3 \cdot 2^{\frac {2}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (7 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 20 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5\right )} e^{\left (\frac {4}{3} i \, d x + \frac {4}{3} i \, c\right )} + 10 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {1}{3}} \log \left (-2 \, \left (\frac {1}{2}\right )^{\frac {2}{3}} a d^{2} \left (-\frac {1}{a d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right ) - 5 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left ({\left (i \, \sqrt {3} a d + a d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (i \, \sqrt {3} a d + a d\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {1}{3}} \log \left (-\left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (i \, \sqrt {3} a d^{2} - a d^{2}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right ) - 5 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left ({\left (-i \, \sqrt {3} a d + a d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-i \, \sqrt {3} a d + a d\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {1}{3}} \log \left (-\left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (-i \, \sqrt {3} a d^{2} - a d^{2}\right )} \left (-\frac {1}{a d^{3}}\right )^{\frac {2}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right )}{20 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\tan ^{3}{\left (c + d x \right )}}{\sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 228, normalized size = 0.96 \begin {gather*} \frac {3}{2\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}+\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{2/3}}{2\,a\,d}-\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/3}}{5\,a^2\,d}+\frac {4^{1/3}\,\ln \left (18\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+9\,4^{2/3}\,{\left (-a\right )}^{1/3}\,d\right )}{4\,{\left (-a\right )}^{1/3}\,d}+\frac {4^{1/3}\,\ln \left (18\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+144\,4^{2/3}\,{\left (-a\right )}^{1/3}\,d\,{\left (-\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}^2\right )\,\left (-\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}{{\left (-a\right )}^{1/3}\,d}-\frac {4^{1/3}\,\ln \left (18\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}+144\,4^{2/3}\,{\left (-a\right )}^{1/3}\,d\,{\left (\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}^2\right )\,\left (\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}{{\left (-a\right )}^{1/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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