Optimal. Leaf size=787 \[ -\frac {4 i b^2 x^{3/2}}{\left (a^2+b^2\right )^2 d}+\frac {4 b^2 x^{3/2}}{(a+i b) (i a+b)^2 d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt {x}\right )}\right )}+\frac {x^2}{2 (a-i b)^2}+\frac {2 b x^2}{(i a-b) (a-i b)^2}-\frac {2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac {6 b^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {4 i b^2 x^{3/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {6 i b^2 \sqrt {x} \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {6 b x \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {6 b^2 x \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {3 b^2 \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {6 b \sqrt {x} \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {6 i b^2 \sqrt {x} \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {3 b \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {3 b^2 \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4} \]
[Out]
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Rubi [A]
time = 1.23, 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 = {3832, 3815,
2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[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*} \int \frac {x}{\left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \, dx &=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 \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) \text {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 ) \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 \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) \text {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 ) \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 \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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 5.55, size = 518, normalized size = 0.66 \begin {gather*} \frac {-\frac {2 i b e^{2 i c} \left (4 b x^2+2 a d x^{5/2}-\frac {e^{-2 i c} \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \sqrt {x} \left (12 b d^2 x \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )+4 a 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 b d^2 x \log \left (-i b \left (-1+e^{2 i \left (c+d \sqrt {x}\right )}\right )+a \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )\right )-6 i \left (b d \sqrt {x}+a d^2 x\right ) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )+3 \left (b+2 a d \sqrt {x}\right ) \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )+3 i a \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )\right )}{(i a+b) d^3}\right )}{d \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right )}+\frac {(a+i b) x^{5/2} (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}+\frac {4 (a+i b) b^2 x^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) (a+i b)^2 \sqrt {x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.99, 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] 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 = 1.04, size = 2477, normalized size = 3.15 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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