Integrand size = 18, antiderivative size = 738 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=-\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1-a) \log (1-a-b x)}{2 b c}+\frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}+\frac {(1+a) \log (1+a+b x)}{2 b c}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3} \]
[Out]
Time = 1.88 (sec) , antiderivative size = 738, 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 = {6251, 196, 46, 2608, 2603, 12, 492, 211, 2605, 457, 78, 2604, 2465, 266, 2463, 2441, 2440, 2438, 214} \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=-\frac {2 \sqrt {a+1} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (-\frac {-a-b x+1}{a+b x}\right ) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d^2 \log \left (\frac {a+b x+1}{a+b x}\right ) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d \sqrt {x} \log \left (-\frac {-a-b x+1}{a+b x}\right )}{c^2}-\frac {d \sqrt {x} \log \left (\frac {a+b x+1}{a+b x}\right )}{c^2}+\frac {(1-a) \log (-a-b x+1)}{2 b c}-\frac {x \log \left (-\frac {-a-b x+1}{a+b x}\right )}{2 c}+\frac {(a+1) \log (a+b x+1)}{2 b c}+\frac {x \log \left (\frac {a+b x+1}{a+b x}\right )}{2 c} \]
[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 6251
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {\log \left (\frac {-1+a+b x}{a+b x}\right )}{c+\frac {d}{\sqrt {x}}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {1+a+b x}{a+b x}\right )}{c+\frac {d}{\sqrt {x}}} \, dx \\ & = -\text {Subst}\left (\int \frac {x^2 \log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \frac {x^2 \log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {d \log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{c^2}+\frac {x \log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{c}+\frac {d^2 \log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \left (-\frac {d \log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{c}+\frac {d^2 \log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\text {Subst}\left (\int x \log \left (\frac {-1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c}+\frac {\text {Subst}\left (\int x \log \left (\frac {1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c}+\frac {d \text {Subst}\left (\int \log \left (\frac {-1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^2}-\frac {d \text {Subst}\left (\int \log \left (\frac {1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^2}-\frac {d^2 \text {Subst}\left (\int \frac {\log \left (\frac {-1+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {d^2 \text {Subst}\left (\int \frac {\log \left (\frac {1+a+b x^2}{a+b x^2}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = \frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {\text {Subst}\left (\int \frac {2 b x^3}{\left (-1+a+b x^2\right ) \left (a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {\text {Subst}\left (\int \frac {2 b x^3}{\left (-a-b x^2\right ) \left (1+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {d \text {Subst}\left (\int \frac {2 b x^2}{\left (-1+a+b x^2\right ) \left (a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {d \text {Subst}\left (\int \frac {2 b x^2}{\left (-a-b x^2\right ) \left (1+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {d^2 \text {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (-\frac {2 b x \left (-1+a+b x^2\right )}{\left (a+b x^2\right )^2}+\frac {2 b x}{a+b x^2}\right ) \log (d+c x)}{-1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3}-\frac {d^2 \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 (1+a+b x^2\right )}{\left (a+b x^2\right )^2}\right ) \log (d+c x)}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3} \\ & = \frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {b \text {Subst}\left (\int \frac {x^3}{\left (-1+a+b x^2\right ) \left (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 (1+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{c}-\frac {(2 b d) \text {Subst}\left (\int \frac {x^2}{\left (-1+a+b x^2\right ) \left (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 (1+a+b x^2\right )} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {d^2 \text {Subst}\left (\int \left (\frac {2 b x \log (d+c x)}{-1+a+b x^2}-\frac {2 b x \log (d+c x)}{a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^3}-\frac {d^2 \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)}{1+a+b x^2}\right ) \, dx,x,\sqrt {x}\right )}{c^3} \\ & = \frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {b \text {Subst}\left (\int \frac {x}{(-1+a+b x) (a+b x)} \, dx,x,x\right )}{2 c}-\frac {b \text {Subst}\left (\int \frac {x}{(-a-b x) (1+a+b x)} \, dx,x,x\right )}{2 c}-\frac {(2 (1-a) d) \text {Subst}\left (\int \frac {1}{-1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 a d) \text {Subst}\left (\int \frac {1}{-a-b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 a d) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {(2 (1+a) d) \text {Subst}\left (\int \frac {1}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \frac {x \log (d+c x)}{-1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3}-\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \frac {x \log (d+c x)}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{c^3} \\ & = -\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}+\frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {b \text {Subst}\left (\int \left (\frac {1-a}{b (-1+a+b x)}+\frac {a}{b (a+b x)}\right ) \, dx,x,x\right )}{2 c}-\frac {b \text {Subst}\left (\int \left (\frac {a}{b (a+b x)}+\frac {-1-a}{b (1+a+b x)}\right ) \, dx,x,x\right )}{2 c}-\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \left (-\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {-1-a}-\sqrt {b} x\right )}+\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {-1-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \left (-\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {1-a}-\sqrt {b} x\right )}+\frac {\log (d+c x)}{2 \sqrt {b} \left (\sqrt {1-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{c^3} \\ & = -\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}+\frac {(1-a) \log (1-a-b x)}{2 b c}+\frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}+\frac {(1+a) \log (1+a+b x)}{2 b c}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {\left (\sqrt {b} d^2\right ) \text {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {-1-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}-\frac {\left (\sqrt {b} d^2\right ) \text {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {1-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}-\frac {\left (\sqrt {b} d^2\right ) \text {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {-1-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {\left (\sqrt {b} d^2\right ) \text {Subst}\left (\int \frac {\log (d+c x)}{\sqrt {1-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{c^3} \\ & = -\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1-a) \log (1-a-b x)}{2 b c}+\frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}+\frac {(1+a) \log (1+a+b x)}{2 b c}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {d^2 \text {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} x\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {d^2 \text {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} x\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {d^2 \text {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} x\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2}-\frac {d^2 \text {Subst}\left (\int \frac {\log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} x\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{d+c x} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = -\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1-a) \log (1-a-b x)}{2 b c}+\frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}+\frac {(1+a) \log (1+a+b x)}{2 b c}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}+\frac {d^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{\sqrt {-1-a} c-\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{\sqrt {1-a} c-\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {-1-a} c+\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {1-a} c+\sqrt {b} d}\right )}{x} \, dx,x,d+c \sqrt {x}\right )}{c^3} \\ & = -\frac {2 \sqrt {1+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1-a) \log (1-a-b x)}{2 b c}+\frac {d \sqrt {x} \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^2}-\frac {x \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 c}-\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {1-a-b x}{a+b x}\right )}{c^3}+\frac {(1+a) \log (1+a+b x)}{2 b c}-\frac {d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )}{c^2}+\frac {x \log \left (\frac {1+a+b x}{a+b x}\right )}{2 c}+\frac {d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 719, normalized size of antiderivative = 0.97 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\frac {-4 \sqrt {1+a} \sqrt {b} c d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )+4 \sqrt {1-a} \sqrt {b} c d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )-2 b d^2 \log \left (\frac {c \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+2 b d^2 \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )-2 b d^2 \log \left (\frac {c \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+2 b d^2 \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+c^2 \log (1-a-b x)-a c^2 \log (1-a-b x)+2 b c d \sqrt {x} \log \left (\frac {-1+a+b x}{a+b x}\right )-b c^2 x \log \left (\frac {-1+a+b x}{a+b x}\right )-2 b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {-1+a+b x}{a+b x}\right )+c^2 \log (1+a+b x)+a c^2 \log (1+a+b x)-2 b c d \sqrt {x} \log \left (\frac {1+a+b x}{a+b x}\right )+b c^2 x \log \left (\frac {1+a+b x}{a+b x}\right )+2 b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {1+a+b x}{a+b x}\right )-2 b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-1-a} c+\sqrt {b} d}\right )-2 b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-1-a} c+\sqrt {b} d}\right )+2 b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {1-a} c+\sqrt {b} d}\right )+2 b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{2 b c^3} \]
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Time = 0.39 (sec) , antiderivative size = 752, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {\operatorname {arccoth}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (-\frac {\left (-1+a \right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b d \left (d +c \sqrt {x}\right )+b \left (d +c \sqrt {x}\right )^{2}-c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 d b +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{2 b}+\frac {\left (1+a \right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b d \left (d +c \sqrt {x}\right )+b \left (d +c \sqrt {x}\right )^{2}+c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 d b +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{2 b}\right )}{2}-c \,d^{2} \left (\frac {-\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\ln \left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{2 b}}{2 c^{2}}+\frac {\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\ln \left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{2 b}}{2 c^{2}}\right )\right )}{c^{2}}\) | \(752\) |
default | \(\frac {\operatorname {arccoth}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (-\frac {\left (-1+a \right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b d \left (d +c \sqrt {x}\right )+b \left (d +c \sqrt {x}\right )^{2}-c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 d b +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{2 b}+\frac {\left (1+a \right ) \left (-\frac {\ln \left (a \,c^{2}+b \,d^{2}-2 b d \left (d +c \sqrt {x}\right )+b \left (d +c \sqrt {x}\right )^{2}+c^{2}\right )}{2 b}+\frac {2 d \arctan \left (\frac {-2 d b +2 b \left (d +c \sqrt {x}\right )}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{\sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{2 b}\right )}{2}-c \,d^{2} \left (\frac {-\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\ln \left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}+b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{2 b}}{2 c^{2}}+\frac {\frac {\ln \left (d +c \sqrt {x}\right ) \left (\ln \left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\ln \left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {d b -b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )+\operatorname {dilog}\left (\frac {-d b +b \left (d +c \sqrt {x}\right )+\sqrt {-a b \,c^{2}-b \,c^{2}}}{-d b +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{2 b}}{2 c^{2}}\right )\right )}{c^{2}}\) | \(752\) |
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+\frac {d}{\sqrt {x}}} \,d x \]
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