Integrand size = 18, antiderivative size = 234 \[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\frac {x^2}{2 (a+i b)}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b x \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 i b \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^4} \]
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Time = 0.36 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3832, 3813, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\frac {3 i b \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 d^4 \left (a^2+b^2\right )}+\frac {3 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^3 \left (a^2+b^2\right )}-\frac {3 i b x \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d \left (a^2+b^2\right )}+\frac {x^2}{2 (a+i b)} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3813
Rule 3832
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^3}{a+b \tan (c+d x)} \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^2}{2 (a+i b)}+(4 i b) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^3}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^2}{2 (a+i b)}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {(6 b) \text {Subst}\left (\int x^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {x^2}{2 (a+i b)}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b x \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {(6 i b) \text {Subst}\left (\int x \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 {x}\right )}{\left (a^2+b^2\right ) d^2} \\ & = \frac {x^2}{2 (a+i b)}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b x \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}-\frac {(3 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d^3} \\ & = \frac {x^2}{2 (a+i b)}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b x \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {(3 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 \left (a^2+b^2\right ) d^4} \\ & = \frac {x^2}{2 (a+i b)}+\frac {2 b x^{3/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i b x \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 i b \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^4} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.91 \[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\frac {a d^4 x^2+i b d^4 x^2+4 b d^3 x^{3/2} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+6 i b d^2 x \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+6 b d \sqrt {x} \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-3 i b \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )}{2 \left (a^2+b^2\right ) d^4} \]
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\[\int \frac {x}{a +b \tan \left (c +d \sqrt {x}\right )}d x\]
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\[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x}{b \tan \left (d \sqrt {x} + c\right ) + a} \,d x } \]
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\[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x}{a + b \tan {\left (c + d \sqrt {x} \right )}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (195) = 390\).
Time = 0.50 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.37 \[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=-\frac {6 \, {\left (\frac {2 \, {\left (d \sqrt {x} + c\right )} a}{a^{2} + b^{2}} + \frac {2 \, b \log \left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (d \sqrt {x} + c\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} c^{3} - \frac {3 \, {\left (d \sqrt {x} + c\right )}^{4} {\left (a - i \, b\right )} - 12 \, {\left (d \sqrt {x} + c\right )}^{3} {\left (a - i \, b\right )} c + 18 \, {\left (d \sqrt {x} + c\right )}^{2} {\left (a - i \, b\right )} c^{2} - 4 \, {\left (4 i \, {\left (d \sqrt {x} + c\right )}^{3} b - 9 i \, {\left (d \sqrt {x} + c\right )}^{2} b c + 9 i \, {\left (d \sqrt {x} + c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}\right ) - 6 \, {\left (4 i \, {\left (d \sqrt {x} + c\right )}^{2} b - 6 i \, {\left (d \sqrt {x} + c\right )} b c + 3 i \, b c^{2}\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}\right ) + 2 \, {\left (4 \, {\left (d \sqrt {x} + c\right )}^{3} b - 9 \, {\left (d \sqrt {x} + c\right )}^{2} b c + 9 \, {\left (d \sqrt {x} + c\right )} b c^{2}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}\right ) + 12 i \, b {\rm Li}_{4}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}) + 6 \, {\left (4 \, {\left (d \sqrt {x} + c\right )} b - 3 \, b c\right )} {\rm Li}_{3}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b})}{a^{2} + b^{2}}}{6 \, d^{4}} \]
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\[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x}{b \tan \left (d \sqrt {x} + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {x}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x}{a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )} \,d x \]
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