Optimal. Leaf size=661 \[ -\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log (-a-b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d^2 \log (a+b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d \sqrt{x} \log (-a-b x+1)}{c^2}-\frac{d \sqrt{x} \log (a+b x+1)}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c} \]
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Rubi [A] time = 1.07218, antiderivative size = 661, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {6115, 2408, 2476, 2448, 321, 205, 2454, 2389, 2295, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ -\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,-\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )}{c^3}+\frac{d^2 \log \left (c \sqrt{x}+d\right ) \log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )}{c^3}-\frac{d^2 \log (-a-b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d^2 \log (a+b x+1) \log \left (c \sqrt{x}+d\right )}{c^3}+\frac{d \sqrt{x} \log (-a-b x+1)}{c^2}-\frac{d \sqrt{x} \log (a+b x+1)}{c^2}-\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} c^2}+\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} c^2}+\frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c} \]
Antiderivative was successfully verified.
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Rule 6115
Rule 2408
Rule 2476
Rule 2448
Rule 321
Rule 205
Rule 2454
Rule 2389
Rule 2295
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rule 208
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx &=-\left (\frac{1}{2} \int \frac{\log (1-a-b x)}{c+\frac{d}{\sqrt{x}}} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+\frac{d}{\sqrt{x}}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x \log \left (1-a-b x^2\right )}{c+\frac{d}{x}} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{x \log \left (1+a+b x^2\right )}{c+\frac{d}{x}} \, dx,x,\sqrt{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{d \log \left (1-a-b x^2\right )}{c^2}+\frac{x \log \left (1-a-b x^2\right )}{c}+\frac{d^2 \log \left (1-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 (1+a+b x^2\right )}{c^2}+\frac{x \log \left (1+a+b x^2\right )}{c}+\frac{d^2 \log \left (1+a+b x^2\right )}{c^2 (d+c x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int x \log \left (1-a-b x^2\right ) \, dx,x,\sqrt{x}\right )}{c}+\frac{\operatorname{Subst}\left (\int x \log \left (1+a+b x^2\right ) \, dx,x,\sqrt{x}\right )}{c}+\frac{d \operatorname{Subst}\left (\int \log \left (1-a-b x^2\right ) \, dx,x,\sqrt{x}\right )}{c^2}-\frac{d \operatorname{Subst}\left (\int \log \left (1+a+b x^2\right ) \, dx,x,\sqrt{x}\right )}{c^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\log \left (1-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 (1+a+b x^2\right )}{d+c x} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=\frac{d \sqrt{x} \log (1-a-b x)}{c^2}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}-\frac{\operatorname{Subst}(\int \log (1-a-b x) \, dx,x,x)}{2 c}+\frac{\operatorname{Subst}(\int \log (1+a+b x) \, dx,x,x)}{2 c}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{x^2}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{c^2}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{x^2}{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{d \sqrt{x} \log (1-a-b x)}{c^2}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{c^3}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-a-b x)}{2 b c}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b 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 (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 \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{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{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{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{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{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{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{d \sqrt{x} \log (1-a-b x)}{c^2}+\frac{(1-a-b x) \log (1-a-b x)}{2 b c}-\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1-a-b x)}{c^3}-\frac{d \sqrt{x} \log (1+a+b x)}{c^2}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}+\frac{d^2 \log \left (d+c \sqrt{x}\right ) \log (1+a+b x)}{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 = 0.606834, size = 598, normalized size = 0.9 \[ \frac{-2 d^2 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{-a-1} c}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )+\log \left (c \sqrt{x}+d\right ) \left (\log \left (\frac{c \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c+\sqrt{b} d}\right )+\log \left (\frac{c \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} c-\sqrt{b} d}\right )\right )\right )+2 d^2 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{b} d-\sqrt{1-a} c}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} \left (c \sqrt{x}+d\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )+\log \left (c \sqrt{x}+d\right ) \left (\log \left (\frac{c \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c+\sqrt{b} d}\right )+\log \left (\frac{c \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} c-\sqrt{b} d}\right )\right )\right )-\frac{c^2 (a+b x-1) \log (-a-b x+1)}{b}+\frac{c^2 (a+b x+1) \log (a+b x+1)}{b}-2 d^2 \log (-a-b x+1) \log \left (c \sqrt{x}+d\right )+2 d^2 \log (a+b x+1) \log \left (c \sqrt{x}+d\right )+2 c d \sqrt{x} \log (-a-b x+1)-2 c d \sqrt{x} \log (a+b x+1)+4 c d \left (\sqrt{x}-\frac{\sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b}}\right )-4 c d \left (\sqrt{x}-\frac{\sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b}}\right )}{2 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 970, normalized size = 1.5 \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*} \int \frac{\operatorname{artanh}\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{c x \operatorname{artanh}\left (b x + a\right ) - d \sqrt{x} \operatorname{artanh}\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{artanh}\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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