Optimal. Leaf size=661 \[ -\frac {d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \text {Li}_2\left (-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )}{c^3}+\frac {d^2 \log \left (c \sqrt {x}+d\right ) \log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )}{c^3}-\frac {d^2 \log (-a-b x+1) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d^2 \log (a+b x+1) \log \left (c \sqrt {x}+d\right )}{c^3}+\frac {d \sqrt {x} \log (-a-b x+1)}{c^2}-\frac {d \sqrt {x} \log (a+b x+1)}{c^2}-\frac {2 \sqrt {a+1} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} c^2}+\frac {2 \sqrt {1-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} c^2}+\frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c} \]
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Rubi [A] time = 1.07, antiderivative size = 661, normalized size of antiderivative = 1.00, 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 205
Rule 208
Rule 260
Rule 321
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2408
Rule 2416
Rule 2448
Rule 2454
Rule 2462
Rule 2476
Rule 6115
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*}
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Mathematica [A] time = 0.64, size = 598, normalized size = 0.90 \[ \frac {-\frac {c^2 (a+b x-1) \log (-a-b x+1)}{b}+\frac {c^2 (a+b x+1) \log (a+b x+1)}{b}-2 d^2 \left (\text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {b} d-\sqrt {-a-1} c}\right )+\text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )+\log \left (c \sqrt {x}+d\right ) \left (\log \left (\frac {c \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c+\sqrt {b} d}\right )+\log \left (\frac {c \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} c-\sqrt {b} d}\right )\right )\right )+2 d^2 \left (\text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {b} d-\sqrt {1-a} c}\right )+\text {Li}_2\left (\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )+\log \left (c \sqrt {x}+d\right ) \left (\log \left (\frac {c \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c+\sqrt {b} d}\right )+\log \left (\frac {c \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} c-\sqrt {b} d}\right )\right )\right )-2 d^2 \log (-a-b x+1) \log \left (c \sqrt {x}+d\right )+2 d^2 \log (a+b x+1) \log \left (c \sqrt {x}+d\right )+2 c d \sqrt {x} \log (-a-b x+1)-2 c d \sqrt {x} \log (a+b x+1)+4 c d \left (\sqrt {x}-\frac {\sqrt {a+1} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b}}\right )-4 c d \left (\sqrt {x}-\frac {\sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b}}\right )}{2 c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {c x \operatorname {artanh}\left (b x + a\right ) - d \sqrt {x} \operatorname {artanh}\left (b x + a\right )}{c^{2} x - d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 970, normalized size = 1.47 \[ \frac {\arctanh \left (b x +a \right ) x}{c}-\frac {2 \arctanh \left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \arctanh \left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {\ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}+c^{2}\right )}{2 b c}-\frac {2 d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}+b \,c^{2}}}+\frac {a \ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}+c^{2}\right )}{2 b c}-\frac {2 a d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}+b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}+b \,c^{2}}}-\frac {a \ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}-c^{2}\right )}{2 b c}+\frac {2 a d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}-b \,c^{2}}}+\frac {\ln \left (b \left (d +c \sqrt {x}\right )^{2}-2 \left (d +c \sqrt {x}\right ) b d +a \,c^{2}+b \,d^{2}-c^{2}\right )}{2 b c}-\frac {2 d \arctan \left (\frac {2 b \left (d +c \sqrt {x}\right )-2 b d}{2 \sqrt {a b \,c^{2}-b \,c^{2}}}\right )}{c \sqrt {a b \,c^{2}-b \,c^{2}}}-\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}-\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}-\frac {d^{2} \dilog \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}-\frac {d^{2} \dilog \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}-b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \dilog \left (\frac {-b \left (d +c \sqrt {x}\right )+b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}}+\frac {d^{2} \dilog \left (\frac {b \left (d +c \sqrt {x}\right )-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}{-b d +\sqrt {-a b \,c^{2}+b \,c^{2}}}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{\sqrt {x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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