Integrand size = 16, antiderivative size = 204 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}+\frac {2 b \left (b+2 a d \sqrt {x}\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 i a b \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \]
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Time = 0.32 (sec) , antiderivative size = 204, 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, 3814, 3813, 2221, 2317, 2438} \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 i a b \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {2 b \left (2 a d \sqrt {x}+b\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {2 b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {\left (2 a d \sqrt {x}+b\right )^2}{2 a d^2 (a+i b) \left (a^2+b^2\right )}-\frac {x}{a^2+b^2} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3813
Rule 3814
Rule 3824
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {x}{a^2+b^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {2 \text {Subst}\left (\int \frac {b+2 a d x}{a+b \tan (c+d x)} \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {(4 i b) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} (b+2 a d x)}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}+\frac {2 b \left (b+2 a d \sqrt {x}\right ) \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 )^2 d^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}-\frac {(4 a b) \text {Subst}\left (\int \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 )^2 d} \\ & = \frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}+\frac {2 b \left (b+2 a d \sqrt {x}\right ) \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 )^2 d^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {(2 i a b) \text {Subst}\left (\int \frac {\log \left (1+\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 )}{\left (a^2+b^2\right )^2 d^2} \\ & = \frac {\left (b+2 a d \sqrt {x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {x}{a^2+b^2}+\frac {2 b \left (b+2 a d \sqrt {x}\right ) \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 )^2 d^2}-\frac {2 i a b \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 )^2 d^2}-\frac {2 b \sqrt {x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(517\) vs. \(2(204)=408\).
Time = 5.67 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.53 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\sec ^2\left (c+d \sqrt {x}\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right ) \left (2 b^2 \left (a^2+b^2\right ) d \sqrt {x} \sin \left (c+d \sqrt {x}\right )-a \left (a^2+b^2\right ) \left (c-d \sqrt {x}\right ) \left (c+d \sqrt {x}\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )-2 b^2 \left (b \left (c+d \sqrt {x}\right )-a \log \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )+4 a b c \left (b \left (c+d \sqrt {x}\right )-a \log \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )-2 a b \left (\sqrt {1+\frac {a^2}{b^2}} b e^{i \arctan \left (\frac {a}{b}\right )} \left (c+d \sqrt {x}\right )^2+a \left (-i \left (c+d \sqrt {x}\right ) \left (\pi -2 \arctan \left (\frac {a}{b}\right )\right )-\pi \log \left (1+e^{-2 i \left (c+d \sqrt {x}\right )}\right )-2 \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right ) \log \left (1-e^{2 i \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right )}\right )+\pi \log \left (\cos \left (c+d \sqrt {x}\right )\right )+2 \arctan \left (\frac {a}{b}\right ) \log \left (\sin \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}+\arctan \left (\frac {a}{b}\right )\right )}\right )\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )\right )}{a \left (a^2+b^2\right )^2 d^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \]
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\[\int \frac {1}{\left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (177) = 354\).
Time = 0.28 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.06 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 \, b^{3} d \sqrt {x} - {\left (a^{3} - a b^{2}\right )} d^{2} x + {\left (a^{3} - a b^{2}\right )} d^{2} - {\left (i \, a b^{2} \tan \left (d \sqrt {x} + c\right ) + i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - {\left (-i \, a b^{2} \tan \left (d \sqrt {x} + c\right ) - i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - 2 \, {\left (a^{2} b d \sqrt {x} + a^{2} b c + {\left (a b^{2} d \sqrt {x} + a b^{2} c\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b d \sqrt {x} + a^{2} b c + {\left (a b^{2} d \sqrt {x} + a b^{2} c\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}}\right ) + {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) + {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) - {\left (2 \, a b^{2} d \sqrt {x} + {\left (a^{2} b - b^{3}\right )} d^{2} x - {\left (a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d^{2} \tan \left (d \sqrt {x} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2}} \]
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\[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (177) = 354\).
Time = 0.60 (sec) , antiderivative size = 994, normalized size of antiderivative = 4.87 \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \]
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