Optimal. Leaf size=830 \[ \frac {i \log \left (\frac {c \left (\sqrt {-a-i}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {i \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {i \log \left (\frac {c \left (\sqrt {-a-i}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {i \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {i \log \left (\sqrt {x} c+d\right ) \log \left (-\frac {-a-b x+i}{a+b x}\right ) d^2}{c^3}-\frac {i \log \left (\sqrt {x} c+d\right ) \log \left (\frac {a+b x+i}{a+b x}\right ) d^2}{c^3}+\frac {i \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) d^2}{c^3}+\frac {i \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {2 i \sqrt {a+i} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right ) d}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right ) d}{\sqrt {b} c^2}-\frac {i \sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right ) d}{c^2}+\frac {i \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right ) d}{c^2}+\frac {(i a+1) \log (-a-b x+i)}{2 b c}+\frac {i x \log \left (-\frac {-a-b x+i}{a+b x}\right )}{2 c}+\frac {(1-i a) \log (a+b x+i)}{2 b c}-\frac {i x \log \left (\frac {a+b x+i}{a+b x}\right )}{2 c} \]
[Out]
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Rubi [A] time = 2.32, antiderivative size = 830, normalized size of antiderivative = 1.00, number of steps used = 65, number of rules used = 19, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {5052, 190, 44, 2528, 2523, 12, 481, 205, 208, 2525, 446, 72, 2524, 2418, 260, 2416, 2394, 2393, 2391} \[ \frac {i \log \left (\frac {c \left (\sqrt {-a-i}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {i \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {i \log \left (\frac {c \left (\sqrt {-a-i}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {i \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {i \log \left (\sqrt {x} c+d\right ) \log \left (-\frac {-a-b x+i}{a+b x}\right ) d^2}{c^3}-\frac {i \log \left (\sqrt {x} c+d\right ) \log \left (\frac {a+b x+i}{a+b x}\right ) d^2}{c^3}+\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) d^2}{c^3}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {2 i \sqrt {a+i} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right ) d}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right ) d}{\sqrt {b} c^2}-\frac {i \sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right ) d}{c^2}+\frac {i \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right ) d}{c^2}+\frac {(i a+1) \log (-a-b x+i)}{2 b c}+\frac {i x \log \left (-\frac {-a-b x+i}{a+b x}\right )}{2 c}+\frac {(1-i a) \log (a+b x+i)}{2 b c}-\frac {i x \log \left (\frac {a+b x+i}{a+b x}\right )}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 72
Rule 190
Rule 205
Rule 208
Rule 260
Rule 446
Rule 481
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2418
Rule 2523
Rule 2524
Rule 2525
Rule 2528
Rule 5052
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx &=\frac {1}{2} i \int \frac {\log \left (\frac {-i+a+b x}{a+b x}\right )}{c+\frac {d}{\sqrt {x}}} \, dx-\frac {1}{2} i \int \frac {\log \left (\frac {i+a+b x}{a+b x}\right )}{c+\frac {d}{\sqrt {x}}} \, dx\\ &=i \operatorname {Subst}\left (\int \frac {x^2 \log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )-i \operatorname {Subst}\left (\int \frac {x^2 \log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )\\ &=i \operatorname {Subst}\left (\int \left (-\frac {d \log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{c^2}+\frac {x \log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{c}+\frac {d^2 \log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt {x}\right )-i \operatorname {Subst}\left (\int \left (-\frac {d \log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{c^2}+\frac {x \log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{c}+\frac {d^2 \log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {i \operatorname {Subst}\left (\int x \log \left (\frac {-i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c}-\frac {i \operatorname {Subst}\left (\int x \log \left (\frac {i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c}-\frac {(i d) \operatorname {Subst}\left (\int \log \left (\frac {-i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^2}+\frac {(i d) \operatorname {Subst}\left (\int \log \left (\frac {i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^2}+\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {-i+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {i+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}-\frac {i \operatorname {Subst}\left (\int \frac {2 i b x^3}{\left (a+b x^2\right ) \left (-i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{2 c}+\frac {i \operatorname {Subst}\left (\int -\frac {2 i b x^3}{\left (a+b x^2\right ) \left (i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{2 c}+\frac {(i d) \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 )}{c^2}-\frac {(i d) \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 )}{c^2}-\frac {\left (i d^2\right ) \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 (d+c x)}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (i d^2\right ) \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 (d+c x)}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}+\frac {b \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x^2\right ) \left (-i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{c}+\frac {b \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x^2\right ) \left (i+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{c}-\frac {(2 b d) \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 )}{c^2}-\frac {(2 b d) \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 )}{c^2}-\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {2 b x \log (d+c x)}{a+b x^2}+\frac {2 b x \log (d+c x)}{-i+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {2 b x \log (d+c x)}{a+b x^2}+\frac {2 b x \log (d+c x)}{i+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^3}\\ &=-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}+\frac {b \operatorname {Subst}\left (\int \frac {x}{(a+b x) (-i+a+b x)} \, dx,x,x\right )}{2 c}+\frac {b \operatorname {Subst}\left (\int \frac {x}{(a+b x) (i+a+b x)} \, dx,x,x\right )}{2 c}-\frac {(2 (1-i a) d) \operatorname {Subst}\left (\int \frac {1}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 (1+i a) d) \operatorname {Subst}\left (\int \frac {1}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {\left (2 i b d^2\right ) \operatorname {Subst}\left (\int \frac {x \log (d+c x)}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (2 i b d^2\right ) \operatorname {Subst}\left (\int \frac {x \log (d+c x)}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=\frac {2 i \sqrt {i+a} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}+\frac {b \operatorname {Subst}\left (\int \left (-\frac {i a}{b (a+b x)}+\frac {1+i a}{b (-i+a+b x)}\right ) \, dx,x,x\right )}{2 c}+\frac {b \operatorname {Subst}\left (\int \left (\frac {i a}{b (a+b x)}+\frac {1-i a}{b (i+a+b x)}\right ) \, dx,x,x\right )}{2 c}+\frac {\left (2 i b d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {-i-a}-\sqrt {b} x\right )}+\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {-i-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{c^3}-\frac {\left (2 i b d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {i-a}-\sqrt {b} x\right )}+\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {i-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{c^3}\\ &=\frac {2 i \sqrt {i+a} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}+\frac {(1+i a) \log (i-a-b x)}{2 b c}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {(1-i a) \log (i+a+b x)}{2 b c}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}-\frac {\left (i \sqrt {b} d^2\right ) \operatorname {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {-i-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (i \sqrt {b} d^2\right ) \operatorname {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {i-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (i \sqrt {b} d^2\right ) \operatorname {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {-i-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}-\frac {\left (i \sqrt {b} d^2\right ) \operatorname {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {i-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=\frac {2 i \sqrt {i+a} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1+i a) \log (i-a-b x)}{2 b c}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {(1-i a) \log (i+a+b x)}{2 b c}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}-\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} x\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} x\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} x\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} x\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=\frac {2 i \sqrt {i+a} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1+i a) \log (i-a-b x)}{2 b c}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {(1-i a) \log (i+a+b x)}{2 b c}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}-\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{\sqrt {-i-a} c-\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}+\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{\sqrt {i-a} c-\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}-\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {-i-a} c+\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}+\frac {\left (i d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {i-a} c+\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}\\ &=\frac {2 i \sqrt {i+a} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1+i a) \log (i-a-b x)}{2 b c}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {(1-i a) \log (i+a+b x)}{2 b c}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}+\frac {i d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )}{c^3}+\frac {i d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 809, normalized size = 0.97 \[ \frac {i a \log (-a-b x+i) c^2+\log (-a-b x+i) c^2+i b x \log \left (\frac {a+b x-i}{a+b x}\right ) c^2-i a \log (a+b x+i) c^2+\log (a+b x+i) c^2-i b x \log \left (\frac {a+b x+i}{a+b x}\right ) c^2+4 i \sqrt {a+i} \sqrt {b} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right ) c-4 i \sqrt {i-a} \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right ) c-2 i b d \sqrt {x} \log \left (\frac {a+b x-i}{a+b x}\right ) c+2 i b d \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right ) c+2 i b d^2 \log \left (\frac {c \left (\sqrt {-a-i}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right )-2 i b d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right )+2 i b d^2 \log \left (\frac {c \left (\sqrt {-a-i}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right )-2 i b d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right )+2 i b d^2 \log \left (\sqrt {x} c+d\right ) \log \left (\frac {a+b x-i}{a+b x}\right )-2 i b d^2 \log \left (\sqrt {x} c+d\right ) \log \left (\frac {a+b x+i}{a+b x}\right )+2 i b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {b} d-\sqrt {-a-i} c}\right )+2 i b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right )-2 i b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {b} d-\sqrt {i-a} c}\right )-2 i b d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{2 b c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {c x \operatorname {arccot}\left (b x + a\right ) - d \sqrt {x} \operatorname {arccot}\left (b x + a\right )}{c^{2} x - d^{2}}, x\right ) \]
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 = 376, normalized size = 0.45 \[ \frac {\mathrm {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\mathrm {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\mathrm {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 c^{2} b a +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+b \,d^{2}}\right )}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 c^{2} b a +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}-5 \textit {\_R}^{2} d +7 \textit {\_R} \,d^{2}-3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{b \,\textit {\_R}^{3}-3 b d \,\textit {\_R}^{2}+a \,c^{2} \textit {\_R} +3 b \,d^{2} \textit {\_R} -a \,c^{2} d -b \,d^{3}}}{2 c} \]
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 \[ \int \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}}\,{d x} \]
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 \[ \int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+\frac {d}{\sqrt {x}}} \,d x \]
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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