Integrand size = 18, antiderivative size = 787 \[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}+\frac {4 b^2 x^{3/2}}{(a+i b) (i a+b)^2 d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {3 b \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {3 b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4} \]
[Out]
Time = 1.90 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3832, 3815, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac {3 b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 b^2 x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac {4 i b^2 x^{3/2}}{d \left (a^2+b^2\right )^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {4 b^2 x^{3/2}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt {x}\right )}+i a-b\right )}-\frac {3 b \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 (-b+i a) (a-i b)^2}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac {2 b x^2}{(-b+i a) (a-i b)^2}+\frac {x^2}{2 (a-i b)^2} \]
[In]
[Out]
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2320
Rule 2611
Rule 3815
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))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^3}{(a-i b)^2}-\frac {4 b^2 x^3}{(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^3}{(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 {x}\right ) \\ & = \frac {x^2}{2 (a-i b)^2}+\frac {(8 b) \text {Subst}\left (\int \frac {x^3}{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 {x}\right )}{(a-i b)^2}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^3}{\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 {x}\right )}{(i a+b)^2} \\ & = \frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^3}{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 {x}\right )}{(i a-b) (a-i b)^2}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^3}{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 {x}\right )}{a^2+b^2}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^3}{\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 {x}\right )}{a^2+b^2} \\ & = -\frac {4 b^2 x^{3/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^3}{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 {x}\right )}{(a+i b)^2 (i a+b)}-\frac {(12 b) \text {Subst}\left (\int x^2 \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 {x}\right )}{(a-i b)^2 (a+i b) d}+\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 {x}\right )}{(a-i b)^2 (a+i b) d} \\ & = -\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{3/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {(12 b) \text {Subst}\left (\int x \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 {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^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 {x}\right )}{(a-i b) (a+i b)^2 d}+\frac {\left (12 i b^2\right ) \text {Subst}\left (\int x^2 \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 {x}\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{3/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {(6 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-\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 {x}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {\left (12 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 {x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int x \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 {x}\right )}{\left (a^2+b^2\right )^2 d^2} \\ & = -\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{3/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {(3 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {\left (6 i 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 {x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (6 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-\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 {x}\right )}{\left (a^2+b^2\right )^2 d^3} \\ & = -\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{3/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {3 b \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {\left (3 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 {x}\right )}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{\left (a^2+b^2\right )^2 d^4} \\ & = -\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}-\frac {4 b^2 x^{3/2}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 x \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {6 b \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {3 b \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {3 b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4} \\ \end{align*}
Time = 2.62 (sec) , antiderivative size = 662, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {-\frac {2 i b \left (4 (a+i b) b (i a+b) d^3 x^{3/2}+2 a (a+i b) (i a+b) d^4 x^2+6 (a-i b) b d^2 \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+4 a (a-i b) d^3 \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x^{3/2} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+3 (a-i b) b \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (2 i d \sqrt {x} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+\operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )\right )+3 a (a-i b) \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (2 i d^2 x \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )+2 d \sqrt {x} \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-i \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )\right )\right )}{d^4 \left (b-b e^{2 i c}-i a \left (1+e^{2 i c}\right )\right )}+\frac {(a-i b)^2 (a+i b) x^2 (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}+\frac {4 (a-i b)^2 (a+i b) b^2 x^{3/2} \sin \left (d \sqrt {x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )}}{2 (a-i b)^3 (a+i b)^2} \]
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\[\int \frac {x}{\left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}}d x\]
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\[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2477 vs. \(2 (638) = 1276\).
Time = 0.90 (sec) , antiderivative size = 2477, normalized size of antiderivative = 3.15 \[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \]
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