Optimal. Leaf size=585 \[ \frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}+\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-1} d}\right )}{d^2}-\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}-\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{d^2}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} d+\sqrt {b} c}\right )}{d^2}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} d+\sqrt {b} c}\right )}{d^2}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}+\frac {c \log (-a-b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log (a+b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log (-a-b x+1)}{d}+\frac {\sqrt {x} \log (a+b x+1)}{d}+\frac {2 \sqrt {a+1} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d} \]
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Rubi [A] time = 1.01, antiderivative size = 585, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6115, 2408, 2466, 2448, 321, 205, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ \frac {c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}+\frac {c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {-a-1} d+\sqrt {b} c}\right )}{d^2}-\frac {c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}-\frac {c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {1-a} d+\sqrt {b} c}\right )}{d^2}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} d+\sqrt {b} c}\right )}{d^2}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} d+\sqrt {b} c}\right )}{d^2}+\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )}{d^2}-\frac {c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}+\frac {c \log (-a-b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log (a+b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log (-a-b x+1)}{d}+\frac {\sqrt {x} \log (a+b x+1)}{d}+\frac {2 \sqrt {a+1} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 260
Rule 321
Rule 2391
Rule 2393
Rule 2394
Rule 2408
Rule 2416
Rule 2448
Rule 2462
Rule 2466
Rule 6115
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx &=-\left (\frac {1}{2} \int \frac {\log (1-a-b x)}{c+d \sqrt {x}} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+d \sqrt {x}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x \log \left (1-a-b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )+\operatorname {Subst}\left (\int \frac {x \log \left (1+a+b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {\log \left (1-a-b x^2\right )}{d}-\frac {c \log \left (1-a-b x^2\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )+\operatorname {Subst}\left (\int \left (\frac {\log \left (1+a+b x^2\right )}{d}-\frac {c \log \left (1+a+b x^2\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \log \left (1-a-b x^2\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\operatorname {Subst}\left (\int \log \left (1+a+b x^2\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {c \operatorname {Subst}\left (\int \frac {\log \left (1-a-b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}-\frac {c \operatorname {Subst}\left (\int \frac {\log \left (1+a+b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {\sqrt {x} \log (1-a-b x)}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)}{d^2}+\frac {\sqrt {x} \log (1+a+b x)}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)}{d^2}+\frac {(2 b c) \operatorname {Subst}\left (\int \frac {x \log (c+d x)}{1-a-b x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 b c) \operatorname {Subst}\left (\int \frac {x \log (c+d x)}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{1-a-b x^2} \, dx,x,\sqrt {x}\right )}{d}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {\sqrt {x} \log (1-a-b x)}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)}{d^2}+\frac {\sqrt {x} \log (1+a+b x)}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)}{d^2}+\frac {(2 b c) \operatorname {Subst}\left (\int \left (-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-1-a}-\sqrt {b} x\right )}+\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-1-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 b c) \operatorname {Subst}\left (\int \left (\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {1-a}-\sqrt {b} x\right )}-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {1-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(2 (1-a)) \operatorname {Subst}\left (\int \frac {1}{1-a-b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 (1+a)) \operatorname {Subst}\left (\int \frac {1}{1+a+b x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 \sqrt {1+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}-\frac {\sqrt {x} \log (1-a-b x)}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)}{d^2}+\frac {\sqrt {x} \log (1+a+b x)}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)}{d^2}-\frac {\left (\sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {-1-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (\sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {1-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (\sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {-1-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (\sqrt {b} c\right ) \operatorname {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {1-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {2 \sqrt {1+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}+\frac {c \log \left (\frac {d \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {c \log \left (-\frac {d \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log (1-a-b x)}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)}{d^2}+\frac {\sqrt {x} \log (1+a+b x)}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)}{d^2}-\frac {c \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {-1-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {c \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}-\frac {c \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {-1-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {-1-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {c \operatorname {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {1-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {1-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 \sqrt {1+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}+\frac {c \log \left (\frac {d \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {c \log \left (-\frac {d \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log (1-a-b x)}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)}{d^2}+\frac {\sqrt {x} \log (1+a+b x)}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)}{d^2}-\frac {c \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {-1-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}-\frac {c \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {-1-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}+\frac {c \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {1-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}+\frac {c \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {1-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}\\ &=\frac {2 \sqrt {1+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1+a}}\right )}{\sqrt {b} d}-\frac {2 \sqrt {1-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b} d}+\frac {c \log \left (\frac {d \left (\sqrt {-1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {c \log \left (-\frac {d \left (\sqrt {-1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {c \log \left (-\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {\sqrt {x} \log (1-a-b x)}{d}+\frac {c \log \left (c+d \sqrt {x}\right ) \log (1-a-b x)}{d^2}+\frac {\sqrt {x} \log (1+a+b x)}{d}-\frac {c \log \left (c+d \sqrt {x}\right ) \log (1+a+b x)}{d^2}+\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-1-a} d}\right )}{d^2}+\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-1-a} d}\right )}{d^2}-\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )}{d^2}-\frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 549, normalized size = 0.94 \[ \frac {c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-1} d}\right )+c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-1} d}\right )-c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {1-a} d}\right )-c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {1-a} d}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {-a-1}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} d+\sqrt {b} c}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {1-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} d+\sqrt {b} c}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {-a-1}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-1} d-\sqrt {b} c}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (\sqrt {1-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {1-a} d-\sqrt {b} c}\right )+c \log (-a-b x+1) \log \left (c+d \sqrt {x}\right )-c \log (a+b x+1) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log (-a-b x+1)+d \sqrt {x} \log (a+b x+1)+\frac {2 \sqrt {a+1} d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+1}}\right )}{\sqrt {b}}-\frac {2 \sqrt {1-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {1-a}}\right )}{\sqrt {b}}}{d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d \sqrt {x} \operatorname {artanh}\left (b x + a\right ) - c \operatorname {artanh}\left (b x + a\right )}{d^{2} x - c^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{d \sqrt {x} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 738, normalized size = 1.26 \[ \frac {2 \arctanh \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctanh \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-b \left (c +d \sqrt {x}\right )+b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}{b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}\right )}{d^{2}}+\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {b \left (c +d \sqrt {x}\right )-b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}{-b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}\right )}{d^{2}}+\frac {c \dilog \left (\frac {-b \left (c +d \sqrt {x}\right )+b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}{b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}\right )}{d^{2}}+\frac {c \dilog \left (\frac {b \left (c +d \sqrt {x}\right )-b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}{-b c +\sqrt {-a \,d^{2} b -b \,d^{2}}}\right )}{d^{2}}-\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-b \left (c +d \sqrt {x}\right )+b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}{b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}\right )}{d^{2}}-\frac {c \ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {b \left (c +d \sqrt {x}\right )-b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}{-b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}\right )}{d^{2}}-\frac {c \dilog \left (\frac {-b \left (c +d \sqrt {x}\right )+b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}{b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}\right )}{d^{2}}-\frac {c \dilog \left (\frac {b \left (c +d \sqrt {x}\right )-b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}{-b c +\sqrt {-a \,d^{2} b +b \,d^{2}}}\right )}{d^{2}}+\frac {2 \arctan \left (\frac {2 b \left (c +d \sqrt {x}\right )-2 b c}{2 \sqrt {a \,d^{2} b +b \,d^{2}}}\right )}{\sqrt {a \,d^{2} b +b \,d^{2}}}+\frac {2 \arctan \left (\frac {2 b \left (c +d \sqrt {x}\right )-2 b c}{2 \sqrt {a \,d^{2} b +b \,d^{2}}}\right ) a}{\sqrt {a \,d^{2} b +b \,d^{2}}}+\frac {2 \arctan \left (\frac {2 b \left (c +d \sqrt {x}\right )-2 b c}{2 \sqrt {a \,d^{2} b -b \,d^{2}}}\right )}{\sqrt {a \,d^{2} b -b \,d^{2}}}-\frac {2 \arctan \left (\frac {2 b \left (c +d \sqrt {x}\right )-2 b c}{2 \sqrt {a \,d^{2} b -b \,d^{2}}}\right ) a}{\sqrt {a \,d^{2} b -b \,d^{2}}} \]
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 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{d \sqrt {x} + c}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+d\,\sqrt {x}} \,d x \]
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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