Optimal. Leaf size=693 \[ -\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c-\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (-\frac {-a-b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right )}{d}-\frac {i \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right )}{d}-\frac {2 i \sqrt {a+i} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {-a+i} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a+i}}\right )}{\sqrt {b} d} \]
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Rubi [A] time = 2.06, antiderivative size = 693, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 16, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {5052, 190, 43, 2528, 2523, 12, 481, 205, 208, 2524, 2418, 260, 2416, 2394, 2393, 2391} \[ -\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a+i} d}\right )}{d^2}+\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c-\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (-\frac {-a-b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right )}{d}-\frac {i \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right )}{d}-\frac {2 i \sqrt {a+i} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {-a+i} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a+i}}\right )}{\sqrt {b} d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 190
Rule 205
Rule 208
Rule 260
Rule 481
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2418
Rule 2523
Rule 2524
Rule 2528
Rule 5052
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx &=\frac {1}{2} i \int \frac {\log \left (\frac {-i+a+b x}{a+b x}\right )}{c+d \sqrt {x}} \, dx-\frac {1}{2} i \int \frac {\log \left (\frac {i+a+b x}{a+b x}\right )}{c+d \sqrt {x}} \, dx\\ &=i \operatorname {Subst}\left (\int \frac {x \log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )-i \operatorname {Subst}\left (\int \frac {x \log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )\\ &=i \operatorname {Subst}\left (\int \left (\frac {\log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{d}-\frac {c \log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )-i \operatorname {Subst}\left (\int \left (\frac {\log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{d}-\frac {c \log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {i \operatorname {Subst}\left (\int \log \left (\frac {-i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {i \operatorname {Subst}\left (\int \log \left (\frac {i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}+\frac {(i c) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (\frac {2 b x}{a+b x^2}-\frac {2 b x \left (-i+a+b x^2\right )}{\left (a+b x^2\right )^2}\right ) \log (c+d x)}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (\frac {2 b x}{a+b x^2}-\frac {2 b x \left (i+a+b x^2\right )}{\left (a+b x^2\right )^2}\right ) \log (c+d x)}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {i \operatorname {Subst}\left (\int \frac {2 i b x^2}{\left (a+b x^2\right ) \left (-i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d}+\frac {i \operatorname {Subst}\left (\int -\frac {2 i b x^2}{\left (a+b x^2\right ) \left (i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}+\frac {(i c) \operatorname {Subst}\left (\int \left (-\frac {2 b x \log (c+d x)}{a+b x^2}+\frac {2 b x \log (c+d x)}{-i+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(i c) \operatorname {Subst}\left (\int \left (-\frac {2 b x \log (c+d x)}{a+b x^2}+\frac {2 b x \log (c+d x)}{i+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right ) \left (-i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right ) \left (i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}+\frac {(2 i b c) \operatorname {Subst}\left (\int \frac {x \log (c+d x)}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(2 i b c) \operatorname {Subst}\left (\int \frac {x \log (c+d x)}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 (1-i a)) \operatorname {Subst}\left (\int \frac {1}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 (1+i a)) \operatorname {Subst}\left (\int \frac {1}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}-\frac {(2 i b c) \operatorname {Subst}\left (\int \left (-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-i-a}-\sqrt {b} x\right )}+\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-i-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 i b c) \operatorname {Subst}\left (\int \left (-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {i-a}-\sqrt {b} x\right )}+\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {i-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}+\frac {\left (i \sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {-i-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (i \sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {i-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (i \sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {-i-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (i \sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {i-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}-\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}+\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {-i-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {-i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {i-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}-\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}+\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {-i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}+\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}\\ &=-\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}-\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 618, normalized size = 0.89 \[ -\frac {i \left (c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )+c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )-c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )-c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{-\sqrt {b} c+\sqrt {-a-i} d}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{-\sqrt {b} c+\sqrt {-a+i} d}\right )+c \log \left (\frac {a+b x-i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log \left (\frac {a+b x-i}{a+b x}\right )+d \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right )+\frac {2 \sqrt {a+i} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b}}-\frac {2 \sqrt {-a+i} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a+i}}\right )}{\sqrt {b}}\right )}{d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d \sqrt {x} \operatorname {arccot}\left (b x + a\right ) - c \operatorname {arccot}\left (b x + a\right )}{d^{2} x - c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 343, normalized size = 0.49 \[ \frac {2 \,\mathrm {arccot}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\mathrm {arccot}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-c \left (\munderset {\textit {\_R1} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 c \,b^{2} \textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+c^{2} b}\right )+\left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 c \,b^{2} \textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{b \,\textit {\_R}^{3}-3 b c \,\textit {\_R}^{2}+a \,d^{2} \textit {\_R} +3 c^{2} b \textit {\_R} -a c \,d^{2}-b \,c^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (b x + a\right )}{d \sqrt {x} + c}\,{d x} \]
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 \[ \int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+d\,\sqrt {x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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