Integrand size = 16, antiderivative size = 610 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=-\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}+\frac {6 b^2 x^{2/3}}{(a+i b) (i a+b)^2 d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3} \]
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Time = 1.59 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3824, 3815, 2216, 2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438} \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=-\frac {3 i b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac {6 i b^2 x^{2/3}}{d \left (a^2+b^2\right )^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x^{2/3}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt [3]{x}\right )}+i a-b\right )}+\frac {3 b \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}+\frac {6 b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac {4 b x}{(-b+i a) (a-i b)^2}+\frac {x}{(a-i b)^2} \]
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Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3815
Rule 3824
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {x^2}{(a-i b)^2}-\frac {4 b^2 x^2}{(i a+b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2}+\frac {4 b x^2}{(a-i b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {x}{(a-i b)^2}+\frac {(12 b) \text {Subst}\left (\int \frac {x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{(i a+b)^2} \\ & = \frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2}-\frac {(12 b) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^2}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2} \\ & = -\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^2}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a+i b)^2 (i a+b)}-\frac {(12 b) \text {Subst}\left (\int x \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d} \\ & = -\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {(6 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b) (a+i b)^2 d}+\frac {\left (12 i b^2\right ) \text {Subst}\left (\int x \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2} \\ & = -\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}+\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (1-\frac {i b}{a}\right ) x}{1+\frac {i b}{a}}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^3} \\ & = -\frac {6 i b^2 x^{2/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{2/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x}{(a-i b)^2}+\frac {4 b x}{(i a-b) (a-i b)^2}-\frac {4 b^2 x}{\left (a^2+b^2\right )^2}+\frac {6 b^2 \sqrt [3]{x} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{2/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3} \\ \end{align*}
Time = 2.91 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\frac {\frac {b \left (\frac {6 b x^{2/3}}{a-i b}+\frac {4 a d x}{a-i b}+\frac {6 b \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \sqrt [3]{x} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )}{(a+i b) (i a+b) d}+\frac {6 a \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x^{2/3} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )}{(a+i b) (i a+b)}+\frac {3 b \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 a \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (2 d \sqrt [3]{x} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-i \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )}{\left (a^2+b^2\right ) d^2}\right )}{d \left (b-b e^{2 i c}-i a \left (1+e^{2 i c}\right )\right )}+\frac {x (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}+\frac {3 b^2 x^{2/3} \sin \left (d \sqrt [3]{x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt [3]{x}\right )+b \sin \left (c+d \sqrt [3]{x}\right )\right )}}{a^2+b^2} \]
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\[\int \frac {1}{{\left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )}^{2}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1187 vs. \(2 (491) = 982\).
Time = 0.27 (sec) , antiderivative size = 1187, normalized size of antiderivative = 1.95 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d \sqrt [3]{x} \right )}\right )^{2}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1732 vs. \(2 (491) = 982\).
Time = 0.70 (sec) , antiderivative size = 1732, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2} \,d x \]
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