Integrand size = 16, antiderivative size = 176 \[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {x}{a+i b}+\frac {3 b x^{2/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^3} \]
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Time = 0.33 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3824, 3813, 2221, 2611, 2320, 6724} \[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {3 b \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d^3 \left (a^2+b^2\right )}-\frac {3 i b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )}+\frac {3 b x^{2/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d \left (a^2+b^2\right )}+\frac {x}{a+i b} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3813
Rule 3824
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{a+b \tan (c+d x)} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {x}{a+i b}+(6 i b) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^2}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {x}{a+i b}+\frac {3 b x^{2/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {(6 b) \text {Subst}\left (\int x \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {x}{a+i b}+\frac {3 b x^{2/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {(3 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^2} \\ & = \frac {x}{a+i b}+\frac {3 b x^{2/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{2 \left (a^2+b^2\right ) d^3} \\ & = \frac {x}{a+i b}+\frac {3 b x^{2/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b \sqrt [3]{x} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^3} \\ \end{align*}
Time = 0.88 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.94 \[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {2 a d^3 x+2 i b d^3 x+6 b d^2 x^{2/3} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+6 i b d \sqrt [3]{x} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+3 b \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )}{2 \left (a^2+b^2\right ) d^3} \]
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\[\int \frac {1}{a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (147) = 294\).
Time = 0.26 (sec) , antiderivative size = 746, normalized size of antiderivative = 4.24 \[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {4 \, a d^{3} x + 6 \, b c^{2} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{\tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 1}\right ) + 6 \, b c^{2} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{\tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 1}\right ) + 6 i \, b d x^{\frac {1}{3}} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - 6 i \, b d x^{\frac {1}{3}} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) + 6 \, {\left (b d^{2} x^{\frac {2}{3}} - b c^{2}\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) + 6 \, {\left (b d^{2} x^{\frac {2}{3}} - b c^{2}\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) + 3 \, b {\rm polylog}\left (3, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} - 2 i \, a b + b^{2} - 2 \, {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right ) + 3 \, b {\rm polylog}\left (3, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} - a^{2} + 2 i \, a b + b^{2} - 2 \, {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + a^{2} + b^{2}}\right )}{4 \, {\left (a^{2} + b^{2}\right )} d^{3}} \]
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\[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\int \frac {1}{a + b \tan {\left (c + d \sqrt [3]{x} \right )}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (147) = 294\).
Time = 0.46 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.53 \[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {3 \, {\left (\frac {2 \, {\left (d x^{\frac {1}{3}} + c\right )} a}{a^{2} + b^{2}} + \frac {2 \, b \log \left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} c^{2} + \frac {2 \, {\left (d x^{\frac {1}{3}} + c\right )}^{3} {\left (a - i \, b\right )} - 6 \, {\left (d x^{\frac {1}{3}} + c\right )}^{2} {\left (a - i \, b\right )} c - 6 \, {\left (i \, {\left (d x^{\frac {1}{3}} + c\right )}^{2} b - 2 i \, {\left (d x^{\frac {1}{3}} + c\right )} b c\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )}{a^{2} + b^{2}}\right ) - 6 \, {\left (i \, {\left (d x^{\frac {1}{3}} + c\right )} b - i \, b c\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d x^{\frac {1}{3}} + 2 i \, c\right )}}{-i \, a + b}\right ) + 3 \, {\left ({\left (d x^{\frac {1}{3}} + c\right )}^{2} b - 2 \, {\left (d x^{\frac {1}{3}} + c\right )} b c\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{\frac {1}{3}} + 2 \, c\right )}{a^{2} + b^{2}}\right ) + 3 \, b {\rm Li}_{3}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d x^{\frac {1}{3}} + 2 i \, c\right )}}{-i \, a + b})}{a^{2} + b^{2}}}{2 \, d^{3}} \]
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\[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\int { \frac {1}{b \tan \left (d x^{\frac {1}{3}} + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {1}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\int \frac {1}{a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )} \,d x \]
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