Integrand size = 16, antiderivative size = 738 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}} \]
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Time = 1.26 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6251, 2593, 2456, 2436, 2332, 2441, 2440, 2438, 199, 327, 211} \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=-\frac {\sqrt {d} \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{2 c^{3/2}}-\frac {\sqrt {d} \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{2 c^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+1)}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+1)}{\sqrt {-c} (1-a)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+1)}{\sqrt {-c} (a+1)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x-1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x-1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac {x \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}+\frac {x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right )}{2 c} \]
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Rule 199
Rule 211
Rule 327
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 2593
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}{x^2}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {1+a+b x}{a+b x}\right )}{c+\frac {d}{x^2}} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\log (-1+a+b x)}{c+\frac {d}{x^2}} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+\frac {d}{x^2}} \, dx-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {1}{c+\frac {d}{x^2}} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {1}{c+\frac {d}{x^2}} \, dx \\ & = -\left (\frac {1}{2} \int \left (\frac {\log (-1+a+b x)}{c}-\frac {d \log (-1+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\log (1+a+b x)}{c}-\frac {d \log (1+a+b x)}{c \left (d+c x^2\right )}\right ) \, dx-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {x^2}{d+c x^2} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {x^2}{d+c x^2} \, dx \\ & = \frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\int \log (-1+a+b x) \, dx}{2 c}+\frac {\int \log (1+a+b x) \, dx}{2 c}+\frac {d \int \frac {\log (-1+a+b x)}{d+c x^2} \, dx}{2 c}-\frac {d \int \frac {\log (1+a+b x)}{d+c x^2} \, dx}{2 c}-\frac {\left (d \left (\log (-1+a+b x)-\log \left (\frac {-1+a+b x}{a+b x}\right )-\log (a+b x)\right )\right ) \int \frac {1}{d+c x^2} \, dx}{2 c}+\frac {\left (d \left (-\log (a+b x)+\log (1+a+b x)-\log \left (\frac {1+a+b x}{a+b x}\right )\right )\right ) \int \frac {1}{d+c x^2} \, dx}{2 c} \\ & = \frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}-\frac {\text {Subst}(\int \log (x) \, dx,x,-1+a+b x)}{2 b c}+\frac {\text {Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}+\frac {d \int \left (\frac {\log (-1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (-1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c}-\frac {d \int \left (\frac {\log (1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (1+a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{2 c} \\ & = \frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \int \frac {\log (-1+a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}+\frac {\sqrt {d} \int \frac {\log (-1+a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c}-\frac {\sqrt {d} \int \frac {\log (1+a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{4 c}-\frac {\sqrt {d} \int \frac {\log (1+a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{4 c} \\ & = \frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(-1+a) \sqrt {-c}+b \sqrt {d}}\right )}{-1+a+b x} \, dx}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}}+\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-\left ((-1+a) \sqrt {-c}\right )+b \sqrt {d}}\right )}{-1+a+b x} \, dx}{4 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-\left ((1+a) \sqrt {-c}\right )+b \sqrt {d}}\right )}{1+a+b x} \, dx}{4 (-c)^{3/2}} \\ & = \frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-\left ((-1+a) \sqrt {-c}\right )+b \sqrt {d}}\right )}{x} \, dx,x,-1+a+b x\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(-1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,-1+a+b x\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-\left ((1+a) \sqrt {-c}\right )+b \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,1+a+b x\right )}{4 (-c)^{3/2}} \\ & = \frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c^{3/2}}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c^{3/2}}+\frac {\sqrt {d} \log (-1+a+b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-1+a+b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.14 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\frac {2 c \log \left (\frac {-1+a+b x}{a+b x}\right )-2 a c \log \left (\frac {-1+a+b x}{a+b x}\right )-2 b c x \log \left (\frac {-1+a+b x}{a+b x}\right )+4 c \log (a+b x)+2 c \log \left (\frac {1+a+b x}{a+b x}\right )+2 a c \log \left (\frac {1+a+b x}{a+b x}\right )+2 b c x \log \left (\frac {1+a+b x}{a+b x}\right )-b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (-1+a+b x)}{-\sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )+b \sqrt {-c} \sqrt {d} \log \left (\frac {-1+a+b x}{a+b x}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )+b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}+b \sqrt {d}}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )-b \sqrt {-c} \sqrt {d} \log \left (\frac {1+a+b x}{a+b x}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )+b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (1-a-b x)}{-\left ((-1+a) \sqrt {-c}\right )+b \sqrt {d}}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )-b \sqrt {-c} \sqrt {d} \log \left (\frac {-1+a+b x}{a+b x}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )-b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}-b \sqrt {d}}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )+b \sqrt {-c} \sqrt {d} \log \left (\frac {1+a+b x}{a+b x}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )-b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{-\sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )+b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{\sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )-b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{\sqrt {-c}+a \sqrt {-c}-b \sqrt {d}}\right )+b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{\sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )}{4 b c^2} \]
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Time = 1.47 (sec) , antiderivative size = 554, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {\ln \left (b x +a +1\right ) x}{2 c}+\frac {\ln \left (b x +a +1\right ) a}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c}-\frac {1}{b c}-\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-c d}-\left (b x +a +1\right ) c +a c +c}{b \sqrt {-c d}+a c +c}\right )}{4 c \sqrt {-c d}}+\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-c d}+\left (b x +a +1\right ) c -a c -c}{b \sqrt {-c d}-a c -c}\right )}{4 c \sqrt {-c d}}-\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}-\left (b x +a +1\right ) c +a c +c}{b \sqrt {-c d}+a c +c}\right )}{4 c \sqrt {-c d}}+\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}+\left (b x +a +1\right ) c -a c -c}{b \sqrt {-c d}-a c -c}\right )}{4 c \sqrt {-c d}}-\frac {\ln \left (b x +a -1\right ) x}{2 c}-\frac {\ln \left (b x +a -1\right ) a}{2 b c}+\frac {\ln \left (b x +a -1\right )}{2 b c}+\frac {d \ln \left (b x +a -1\right ) \ln \left (\frac {b \sqrt {-c d}-\left (b x +a -1\right ) c +a c -c}{b \sqrt {-c d}+a c -c}\right )}{4 c \sqrt {-c d}}-\frac {d \ln \left (b x +a -1\right ) \ln \left (\frac {b \sqrt {-c d}+\left (b x +a -1\right ) c -a c +c}{b \sqrt {-c d}-a c +c}\right )}{4 c \sqrt {-c d}}+\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}-\left (b x +a -1\right ) c +a c -c}{b \sqrt {-c d}+a c -c}\right )}{4 c \sqrt {-c d}}-\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}+\left (b x +a -1\right ) c -a c +c}{b \sqrt {-c d}-a c +c}\right )}{4 c \sqrt {-c d}}\) | \(554\) |
derivativedivides | \(\text {Expression too large to display}\) | \(9136\) |
default | \(\text {Expression too large to display}\) | \(9136\) |
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 651, normalized size of antiderivative = 0.88 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=-{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \operatorname {arcoth}\left (b x + a\right ) + \frac {2 \, {\left (a + 1\right )} c \log \left (b x + a + 1\right ) - 2 \, {\left (a - 1\right )} c \log \left (b x + a - 1\right ) + {\left (b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (\frac {{\left (a - 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a + i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} - 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (-\frac {{\left (a - 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a + i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (\frac {{\left (a + 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a - i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} + 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (-\frac {{\left (a + 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a - i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - {\left (b \arctan \left (\frac {{\left (b^{2} x + {\left (a + 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}, \frac {{\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {{\left (b^{2} x + {\left (a - 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}, \frac {{\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right )\right )} \log \left (c x^{2} + d\right )\right )} \sqrt {c} \sqrt {d}}{4 \, b c^{2}} \]
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \]
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