Optimal. Leaf size=738 \[ -\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (-\frac{-a-b x+1}{a+b x}\right ) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d^2 \log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d \sqrt{x} \log \left (-\frac{-a-b x+1}{a+b x}\right )}{c^2}-\frac{d \sqrt{x} \log \left (\frac{a+b x+1}{a+b x}\right )}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{(1-a) \log (-a-b x+1)}{2 b c}-\frac{x \log \left (-\frac{-a-b x+1}{a+b x}\right )}{2 c}+\frac{(a+1) \log (a+b x+1)}{2 b c}+\frac{x \log \left (\frac{a+b x+1}{a+b x}\right )}{2 c} \]
[Out]
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Rubi [A] time = 2.37149, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 65, number of rules used = 19, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.056, Rules used = {6116, 190, 44, 2528, 2523, 12, 481, 205, 2525, 446, 72, 2524, 2418, 260, 2416, 2394, 2393, 2391, 208} \[ -\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (-\frac{-a-b x+1}{a+b x}\right ) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d^2 \log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d \sqrt{x} \log \left (-\frac{-a-b x+1}{a+b x}\right )}{c^2}-\frac{d \sqrt{x} \log \left (\frac{a+b x+1}{a+b x}\right )}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{(1-a) \log (-a-b x+1)}{2 b c}-\frac{x \log \left (-\frac{-a-b x+1}{a+b x}\right )}{2 c}+\frac{(a+1) \log (a+b x+1)}{2 b c}+\frac{x \log \left (\frac{a+b x+1}{a+b x}\right )}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6116
Rule 190
Rule 44
Rule 2528
Rule 2523
Rule 12
Rule 481
Rule 205
Rule 2525
Rule 446
Rule 72
Rule 2524
Rule 2418
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rule 208
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx &=-\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\\ &=-\operatorname{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 )+\operatorname{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 )\\ &=-\operatorname{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 )+\operatorname{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{\operatorname{Subst}\left (\int x \log \left (\frac{-1+a+b x^2}{a+b x^2}\right ) \, dx,x,\sqrt{x}\right )}{c}+\frac{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right ) \left (\frac{2 b x}{a+b x^2}-\frac{2 b x \left (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 \operatorname{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 \operatorname{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) \operatorname{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) \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \left (-\frac{2 b x \log (d+c x)}{a+b x^2}+\frac{2 b x \log (d+c x)}{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 \operatorname{Subst}\left (\int \frac{x}{(-1+a+b x) (a+b x)} \, dx,x,x\right )}{2 c}-\frac{b \operatorname{Subst}\left (\int \frac{x}{(-a-b x) (1+a+b x)} \, dx,x,x\right )}{2 c}-\frac{(2 (1-a) d) \operatorname{Subst}\left (\int \frac{1}{-1+a+b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{(2 a d) \operatorname{Subst}\left (\int \frac{1}{-a-b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{(2 a d) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{c^2}-\frac{(2 (1+a) d) \operatorname{Subst}\left (\int \frac{1}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{\left (2 b d^2\right ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\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{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 \operatorname{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 \operatorname{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 \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\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{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 \operatorname{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 \operatorname{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 \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\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{Li}_2\left (-\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{-1-a} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{Li}_2\left (-\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{Li}_2\left (\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{-1-a} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{Li}_2\left (\frac{\sqrt{b} \left (d+c \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}\\ \end{align*}
Mathematica [A] time = 0.693836, size = 719, normalized size = 0.97 \[ \frac{-2 b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{-a-1} c}\right )-2 b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )+2 b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{1-a} c}\right )+2 b d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )-a c^2 \log (-a-b x+1)+c^2 \log (-a-b x+1)-b c^2 x \log \left (\frac{a+b x-1}{a+b x}\right )+a c^2 \log (a+b x+1)+c^2 \log (a+b x+1)+b c^2 x \log \left (\frac{a+b x+1}{a+b x}\right )-2 b d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )+2 b d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )-2 b d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )+2 b d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )-2 b d^2 \log \left (\frac{a+b x-1}{a+b x}\right ) \log \left (c \sqrt{x}+d\right )+2 b d^2 \log \left (\frac{a+b x+1}{a+b x}\right ) \log \left (c \sqrt{x}+d\right )+2 b c d \sqrt{x} \log \left (\frac{a+b x-1}{a+b x}\right )-2 b c d \sqrt{x} \log \left (\frac{a+b x+1}{a+b x}\right )-4 \sqrt{a+1} \sqrt{b} c d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )+4 \sqrt{1-a} \sqrt{b} c d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{2 b c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 970, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b x + a + 1\right )} \log \left (b x + a + 1\right ) -{\left (b x + a - 1\right )} \log \left (b x + a - 1\right )}{2 \, b c} - \frac{1}{2} \, \int \frac{d \log \left (b x + a + 1\right ) - d \log \left (b x + a - 1\right )}{c^{2} \sqrt{x} + c d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{c x \operatorname{arcoth}\left (b x + a\right ) - d \sqrt{x} \operatorname{arcoth}\left (b x + a\right )}{c^{2} x - d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (b x + a\right )}{c + \frac{d}{\sqrt{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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