Optimal. Leaf size=787 \[ \frac {3 b^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac {3 b^2 \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 i b^2 \sqrt {x} \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 b^2 x \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac {4 i b^2 x^{3/2}}{d \left (a^2+b^2\right )^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {4 b^2 x^{3/2}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt {x}\right )}+i a-b\right )}-\frac {3 b \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 (-b+i a) (a-i b)^2}+\frac {6 b \sqrt {x} \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}+\frac {6 b x \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac {2 b x^2}{(-b+i a) (a-i b)^2}+\frac {x^2}{2 (a-i b)^2} \]
[Out]
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Rubi [A] time = 1.71, antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3747, 3734, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \[ -\frac {6 b^2 x \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}-\frac {6 i b^2 \sqrt {x} \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 \left (a^2+b^2\right )^2}+\frac {3 b^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac {3 b^2 \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 \left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d \left (a^2+b^2\right )^2}-\frac {4 i b^2 x^{3/2}}{d \left (a^2+b^2\right )^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {4 b^2 x^{3/2}}{d (a+i b) (b+i a)^2 \left ((b+i a) e^{2 i \left (c+d \sqrt {x}\right )}+i a-b\right )}+\frac {6 b x \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^2 (-b+i a) (a-i b)^2}+\frac {6 b \sqrt {x} \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^3 (a-i b)^2 (a+i b)}-\frac {3 b \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d^4 (-b+i a) (a-i b)^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{d (a-i b)^2 (a+i b)}+\frac {2 b x^2}{(-b+i a) (a-i b)^2}+\frac {x^2}{2 (a-i b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2282
Rule 2531
Rule 3734
Rule 3747
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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 ) \operatorname {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 \text {Li}_2\left (-\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) \operatorname {Subst}\left (\int x \text {Li}_2\left (-\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 ) \operatorname {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 ) \operatorname {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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_3\left (-\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) \operatorname {Subst}\left (\int \text {Li}_3\left (-\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 ) \operatorname {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 ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 {Subst}\left (\int \frac {\text {Li}_3\left (-\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 ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\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 ) \operatorname {Subst}\left (\int \text {Li}_3\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 {Li}_4\left (-\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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 {Li}_4\left (-\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 \text {Li}_4\left (-\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*}
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Mathematica [A] time = 4.52, size = 633, normalized size = 0.80 \[ \frac {\frac {2 b \left (\frac {3 b \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \left (2 d \sqrt {x} \text {Li}_2\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-i \text {Li}_3\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )\right )}{d^3 \left (a^2+b^2\right )}+\frac {3 a \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \left (2 d^2 x \text {Li}_2\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-2 i d \sqrt {x} \text {Li}_3\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )-\text {Li}_4\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )\right )}{d^3 \left (a^2+b^2\right )}+\frac {4 a x^{3/2} \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )}{(a+i b) (b+i a)}+\frac {6 b x \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt {x}\right )}}{a-i b}\right )}{d (a+i b) (b+i a)}+\frac {2 a d x^2}{a-i b}+\frac {4 b x^{3/2}}{a-i b}\right )}{d \left (-i a \left (1+e^{2 i c}\right )+b \left (-e^{2 i c}\right )+b\right )}+\frac {4 b^2 x^{3/2} \sin \left (d \sqrt {x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )}+\frac {x^2 (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}}{2 \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{b^{2} \tan \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \tan \left (d \sqrt {x} + c\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.31, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.31, size = 2484, normalized size = 3.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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