Integrand size = 21, antiderivative size = 318 \[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {1}{4} i \sqrt [3]{c-i d} x-\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}+\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f} \]
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Time = 0.50 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3609, 3620, 3618, 59, 631, 210, 31} \[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}+\frac {3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}+\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}+\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac {1}{4} i x \sqrt [3]{c-i d}-\frac {1}{4} i x \sqrt [3]{c+i d} \]
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Rule 31
Rule 59
Rule 210
Rule 631
Rule 3609
Rule 3618
Rule 3620
Rubi steps \begin{align*} \text {integral}& = \frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\int \frac {-d+c \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx \\ & = \frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {1}{2} (-i c-d) \int \frac {1+i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx+\frac {1}{2} (i c-d) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx \\ & = \frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {(c-i d) \text {Subst}\left (\int \frac {1}{(-1+x) (c-i d x)^{2/3}} \, dx,x,i \tan (e+f x)\right )}{2 f}+\frac {(c+i d) \text {Subst}\left (\int \frac {1}{(-1+x) (c+i d x)^{2/3}} \, dx,x,-i \tan (e+f x)\right )}{2 f} \\ & = \frac {1}{4} i \sqrt [3]{c-i d} x-\frac {1}{4} i \sqrt [3]{c+i d} x+\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}+\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac {\left (3 \sqrt [3]{c-i d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{c-i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {\left (3 (c-i d)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{(c-i d)^{2/3}+\sqrt [3]{c-i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {\left (3 \sqrt [3]{c+i d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{c+i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {\left (3 (c+i d)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{(c+i d)^{2/3}+\sqrt [3]{c+i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f} \\ & = \frac {1}{4} i \sqrt [3]{c-i d} x-\frac {1}{4} i \sqrt [3]{c+i d} x+\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}+\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {\left (3 \sqrt [3]{c-i d}\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-i d}}\right )}{2 f}+\frac {\left (3 \sqrt [3]{c+i d}\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+i d}}\right )}{2 f} \\ & = \frac {1}{4} i \sqrt [3]{c-i d} x-\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}+\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.09 \[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {2 \sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )-2 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )-2 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\sqrt [3]{c-i d} \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+\sqrt [3]{c+i d} \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )-12 \sqrt [3]{c+d \tan (e+f x)}}{4 f} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.51 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.28
method | result | size |
derivativedivides | \(\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{3} c -c^{2}-d^{2}\right ) \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2}}{f}\) | \(88\) |
default | \(\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{3} c -c^{2}-d^{2}\right ) \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2}}{f}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (234) = 468\).
Time = 0.29 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.50 \[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {{\left (\sqrt {-3} f + f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} f + f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} f - f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} f - f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 2 \, f \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-f \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + {\left (\sqrt {-3} f + f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} f + f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} f - f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} f - f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 2 \, f \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-f \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}{4 \, f} \]
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\[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\, dx \]
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\[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \tan \left (f x + e\right ) \,d x } \]
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Time = 2.60 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.05 \[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {3 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}{f} \]
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Time = 9.02 (sec) , antiderivative size = 830, normalized size of antiderivative = 2.61 \[ \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \]
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