Integrand size = 18, antiderivative size = 693 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=-\frac {2 i \sqrt {i+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}-\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2} \]
[Out]
Time = 1.53 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {5160, 196, 45, 2608, 2603, 12, 492, 211, 214, 2604, 2465, 266, 2463, 2441, 2440, 2438} \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=-\frac {2 i \sqrt {a+i} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {-a+i} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a+i}}\right )}{\sqrt {b} d}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c-\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log \left (-\frac {-a-b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right )}{d}-\frac {i \sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right )}{d} \]
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Rule 12
Rule 45
Rule 196
Rule 211
Rule 214
Rule 266
Rule 492
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2465
Rule 2603
Rule 2604
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+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 \text {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 \text {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 \text {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 \text {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 \text {Subst}\left (\int \log \left (\frac {-i+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {i \text {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) \text {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) \text {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) \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 (c+d x)}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(i c) \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 (c+d x)}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {i \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 )}{d}+\frac {i \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 )}{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) \text {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) \text {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) \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 )}{d}+\frac {(2 b) \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 )}{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) \text {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) \text {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)) \text {Subst}\left (\int \frac {1}{i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 (1+i a)) \text {Subst}\left (\int \frac {1}{-i+a+b x^2} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i \sqrt {i+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\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) \text {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) \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\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 {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) \text {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) \text {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) \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\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 {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) \text {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) \text {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) \text {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} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\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 {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 618, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=-\frac {i \left (\frac {2 \sqrt {i+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b}}-\frac {2 \sqrt {i-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b}}+c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )+c \log \left (\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log \left (\frac {-i+a+b x}{a+b x}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {-i+a+b x}{a+b x}\right )+d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )\right )}{d^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.43 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {arccot}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\) | \(364\) |
default | \(\frac {2 \,\operatorname {arccot}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\) | \(364\) |
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\[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d \sqrt {x} + c} \,d x } \]
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Exception generated. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+d\,\sqrt {x}} \,d x \]
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