Integrand size = 18, antiderivative size = 830 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\frac {2 i \sqrt {i+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1+i a) \log (i-a-b x)}{2 b c}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {(1-i a) \log (i+a+b x)}{2 b c}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}+\frac {i d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )}{c^3}+\frac {i d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{c^3} \]
[Out]
Time = 1.73 (sec) , antiderivative size = 830, normalized size of antiderivative = 1.00, number of steps used = 65, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {5160, 196, 46, 2608, 2603, 12, 492, 211, 214, 2605, 457, 78, 2604, 2465, 266, 2463, 2441, 2440, 2438} \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\frac {i \log \left (\frac {c \left (\sqrt {-a-i}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {i \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {i \log \left (\frac {c \left (\sqrt {-a-i}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {i \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {i \log \left (\sqrt {x} c+d\right ) \log \left (-\frac {-a-b x+i}{a+b x}\right ) d^2}{c^3}-\frac {i \log \left (\sqrt {x} c+d\right ) \log \left (\frac {a+b x+i}{a+b x}\right ) d^2}{c^3}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) d^2}{c^3}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) d^2}{c^3}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {2 i \sqrt {a+i} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right ) d}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right ) d}{\sqrt {b} c^2}-\frac {i \sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right ) d}{c^2}+\frac {i \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right ) d}{c^2}+\frac {(i a+1) \log (-a-b x+i)}{2 b c}+\frac {i x \log \left (-\frac {-a-b x+i}{a+b x}\right )}{2 c}+\frac {(1-i a) \log (a+b x+i)}{2 b c}-\frac {i x \log \left (\frac {a+b x+i}{a+b x}\right )}{2 c} \]
[In]
[Out]
Rule 12
Rule 46
Rule 78
Rule 196
Rule 211
Rule 214
Rule 266
Rule 457
Rule 492
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2465
Rule 2603
Rule 2604
Rule 2605
Rule 2608
Rule 5160
Rubi steps \begin{align*} \text {integral}& = \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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 \text {Subst}\left (\int \frac {x}{(a+b x) (-i+a+b x)} \, dx,x,x\right )}{2 c}+\frac {b \text {Subst}\left (\int \frac {x}{(a+b x) (i+a+b x)} \, dx,x,x\right )}{2 c}-\frac {(2 (1-i a) d) \text {Subst}\left (\int \frac {1}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 (1+i a) d) \text {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 ) \text {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 ) \text {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 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\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 \text {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 \text {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 ) \text {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 ) \text {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 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\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 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )}{c^3}+\frac {i d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{c^3} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 809, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\frac {4 i \sqrt {i+a} \sqrt {b} c d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )-4 i \sqrt {i-a} \sqrt {b} c d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )+2 i b d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )-2 i b d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+2 i b d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )-2 i b d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+c^2 \log (i-a-b x)+i a c^2 \log (i-a-b x)-2 i b c d \sqrt {x} \log \left (\frac {-i+a+b x}{a+b x}\right )+i b c^2 x \log \left (\frac {-i+a+b x}{a+b x}\right )+2 i b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {-i+a+b x}{a+b x}\right )+c^2 \log (i+a+b x)-i a c^2 \log (i+a+b x)+2 i b c d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )-i b c^2 x \log \left (\frac {i+a+b x}{a+b x}\right )-2 i b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )+2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-i-a} c+\sqrt {b} d}\right )+2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )-2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {i-a} c+\sqrt {b} d}\right )-2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{2 b c^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.47 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.47
method | result | size |
derivativedivides | \(\frac {\operatorname {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+5 \textit {\_R}^{2} d -7 \textit {\_R} \,d^{2}+3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b d +\textit {\_R} a \,c^{2}+3 \textit {\_R} b \,d^{2}-a \,c^{2} d -b \,d^{3}}\right )}{8 b}+\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+b \,d^{2}}\right )}{4 b}\right )}{c^{2}}\) | \(388\) |
default | \(\frac {\operatorname {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+5 \textit {\_R}^{2} d -7 \textit {\_R} \,d^{2}+3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b d +\textit {\_R} a \,c^{2}+3 \textit {\_R} b \,d^{2}-a \,c^{2} d -b \,d^{3}}\right )}{8 b}+\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+b \,d^{2}}\right )}{4 b}\right )}{c^{2}}\) | \(388\) |
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\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
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Exception generated. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+\frac {d}{\sqrt {x}}} \,d x \]
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