Optimal. Leaf size=546 \[ \frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {d \text {ArcTan}\left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} f}-\frac {\sqrt {3} d \sqrt [3]{c-\sqrt {-d^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {\sqrt {3} d \sqrt [3]{c+\sqrt {-d^2}} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {d \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}+\frac {3 d \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f} \]
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Rubi [A]
time = 0.45, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3649, 3734,
3566, 726, 52, 59, 631, 210, 31, 3715} \begin {gather*} -\frac {d \text {ArcTan}\left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} f}-\frac {\sqrt {3} d \sqrt [3]{c-\sqrt {-d^2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {\sqrt {3} d \sqrt [3]{c+\sqrt {-d^2}} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}+\frac {3 d \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}+\frac {d \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}+\frac {1}{4} x \sqrt [3]{c-\sqrt {-d^2}}+\frac {1}{4} x \sqrt [3]{c+\sqrt {-d^2}}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 631
Rule 726
Rule 3566
Rule 3649
Rule 3715
Rule 3734
Rubi steps
\begin {align*} \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx &=-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}-\int \frac {\cot (e+f x) \left (-\frac {d}{3}+c \tan (e+f x)+\frac {2}{3} d \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{2/3}} \, dx\\ &=-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {1}{3} d \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{2/3}} \, dx-\int \sqrt [3]{c+d \tan (e+f x)} \, dx\\ &=-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {d \text {Subst}\left (\int \frac {1}{x (c+d x)^{2/3}} \, dx,x,\tan (e+f x)\right )}{3 f}-\frac {d \text {Subst}\left (\int \frac {\sqrt [3]{c+x}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac {d \text {Subst}\left (\int \left (\frac {\sqrt {-d^2} \sqrt [3]{c+x}}{2 d^2 \left (\sqrt {-d^2}-x\right )}+\frac {\sqrt {-d^2} \sqrt [3]{c+x}}{2 d^2 \left (\sqrt {-d^2}+x\right )}\right ) \, dx,x,d \tan (e+f x)\right )}{f}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac {d \text {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{2 \sqrt [3]{c} f}\\ &=-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {d \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}\right )}{c^{2/3} f}+\frac {d \text {Subst}\left (\int \frac {\sqrt [3]{c+x}}{\sqrt {-d^2}-x} \, dx,x,d \tan (e+f x)\right )}{2 \sqrt {-d^2} f}+\frac {d \text {Subst}\left (\int \frac {\sqrt [3]{c+x}}{\sqrt {-d^2}+x} \, dx,x,d \tan (e+f x)\right )}{2 \sqrt {-d^2} f}\\ &=-\frac {d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {\left (d \left (c+\sqrt {-d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d^2}-x\right ) (c+x)^{2/3}} \, dx,x,d \tan (e+f x)\right )}{2 \sqrt {-d^2} f}-\frac {\left (d^2+c \sqrt {-d^2}\right ) \text {Subst}\left (\int \frac {1}{(c+x)^{2/3} \left (\sqrt {-d^2}+x\right )} \, dx,x,d \tan (e+f x)\right )}{2 d f}\\ &=\frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} f}-\frac {\sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {\left (3 d \sqrt [3]{c+\sqrt {-d^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{c+\sqrt {-d^2}}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}+\frac {\left (3 d \left (c+\sqrt {-d^2}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (c+\sqrt {-d^2}\right )^{2/3}+\sqrt [3]{c+\sqrt {-d^2}} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}+\frac {\left (3 \left (d^2+c \sqrt {-d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{c-\sqrt {-d^2}}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d \left (c-\sqrt {-d^2}\right )^{2/3} f}+\frac {\left (3 \left (d^2+c \sqrt {-d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (c-\sqrt {-d^2}\right )^{2/3}+\sqrt [3]{c-\sqrt {-d^2}} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d \sqrt [3]{c-\sqrt {-d^2}} f}\\ &=\frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} f}-\frac {\sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac {3 \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac {\left (3 d \sqrt [3]{c+\sqrt {-d^2}}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}\right )}{2 \sqrt {-d^2} f}-\frac {\left (3 \left (d^2+c \sqrt {-d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}\right )}{2 d \left (c-\sqrt {-d^2}\right )^{2/3} f}\\ &=\frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} f}+\frac {\sqrt {3} \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 d f}+\frac {\sqrt {3} d \sqrt [3]{c+\sqrt {-d^2}} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}-\frac {\sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 d f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}-\frac {3 \sqrt {-d^2} \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 d f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.40, size = 464, normalized size = 0.85 \begin {gather*} \frac {\frac {1}{3} \sqrt [3]{c} d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )-\frac {1}{6} \sqrt [3]{c} d \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )+\log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right )+\frac {1}{4} i c \sqrt [3]{c-i d} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right )-\frac {1}{4} i c \sqrt [3]{c+i d} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right )+d \sqrt [3]{c+d \tan (e+f x)}-\cot (e+f x) (c+d \tan (e+f x))^{4/3}}{c f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \left (\cot ^{2}\left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \cot ^{2}{\left (e + f x \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 = 21.08, size = 2500, normalized size = 4.58 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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