Optimal. Leaf size=470 \[ -\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {a e g \log \left (f+g x^2\right )}{2 f}+\frac {a e g \log (x)}{f}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 f}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 f}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 f}+\frac {b e \left (c^2 f+g\right ) \log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)}{f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {b e g \text {Li}_2(-c x)}{2 f}+\frac {b e g \text {Li}_2(c x)}{2 f}+\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}} \]
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Rubi [A] time = 0.75, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 17, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5916, 325, 206, 6085, 801, 635, 205, 260, 446, 72, 6725, 5912, 5992, 5920, 2402, 2315, 2447} \[ -\frac {b e \left (c^2 f+g\right ) \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 f}+\frac {b e \left (c^2 f+g\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{4 f}+\frac {b e \left (c^2 f+g\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{4 f}-\frac {b e g \text {PolyLog}(2,-c x)}{2 f}+\frac {b e g \text {PolyLog}(2,c x)}{2 f}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {a e g \log \left (f+g x^2\right )}{2 f}+\frac {a e g \log (x)}{f}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e \left (c^2 f+g\right ) \log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)}{f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}} \]
Antiderivative was successfully verified.
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Rule 72
Rule 205
Rule 206
Rule 260
Rule 325
Rule 446
Rule 635
Rule 801
Rule 2315
Rule 2402
Rule 2447
Rule 5912
Rule 5916
Rule 5920
Rule 5992
Rule 6085
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(2 e g) \int \left (\frac {-a-b c x}{2 x \left (f+g x^2\right )}+\frac {b \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)}{2 x \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \frac {-a-b c x}{x \left (f+g x^2\right )} \, dx-(b e g) \int \frac {\left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx\\ &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \left (-\frac {a}{f x}+\frac {-b c f+a g x}{f \left (f+g x^2\right )}\right ) \, dx-(b e g) \int \left (-\frac {\tanh ^{-1}(c x)}{f x}+\frac {\left (c^2 f+g\right ) x \tanh ^{-1}(c x)}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {a e g \log (x)}{f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {(e g) \int \frac {-b c f+a g x}{f+g x^2} \, dx}{f}+\frac {(b e g) \int \frac {\tanh ^{-1}(c x)}{x} \, dx}{f}-\frac {\left (b e g \left (c^2 f+g\right )\right ) \int \frac {x \tanh ^{-1}(c x)}{f+g x^2} \, dx}{f}\\ &=\frac {a e g \log (x)}{f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {b e g \text {Li}_2(-c x)}{2 f}+\frac {b e g \text {Li}_2(c x)}{2 f}+(b c e g) \int \frac {1}{f+g x^2} \, dx-\frac {\left (a e g^2\right ) \int \frac {x}{f+g x^2} \, dx}{f}-\frac {\left (b e g \left (c^2 f+g\right )\right ) \int \left (-\frac {\tanh ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\tanh ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {b e g \text {Li}_2(-c x)}{2 f}+\frac {b e g \text {Li}_2(c x)}{2 f}+\frac {\left (b e \sqrt {g} \left (c^2 f+g\right )\right ) \int \frac {\tanh ^{-1}(c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f}-\frac {\left (b e \sqrt {g} \left (c^2 f+g\right )\right ) \int \frac {\tanh ^{-1}(c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {b e g \text {Li}_2(-c x)}{2 f}+\frac {b e g \text {Li}_2(c x)}{2 f}-2 \frac {\left (b c e \left (c^2 f+g\right )\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac {\left (b c e \left (c^2 f+g\right )\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac {\left (b c e \left (c^2 f+g\right )\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {b e g \text {Li}_2(-c x)}{2 f}+\frac {b e g \text {Li}_2(c x)}{2 f}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f}-2 \frac {\left (b e \left (c^2 f+g\right )\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {b e g \text {Li}_2(-c x)}{2 f}+\frac {b e g \text {Li}_2(c x)}{2 f}-\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 f}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f}\\ \end {align*}
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Mathematica [C] time = 4.97, size = 982, normalized size = 2.09 \[ -\frac {-4 b e g \tanh ^{-1}(c x)^2 x^2-4 b c e \sqrt {f} \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) x^2-2 b c^2 d f \tanh ^{-1}(c x) x^2-4 i b c^2 e f \sin ^{-1}\left (\sqrt {\frac {c^2 f}{f c^2+g}}\right ) \tanh ^{-1}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right ) x^2-4 b e g \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right ) x^2-4 b c^2 e f \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right ) x^2-2 i b c^2 e f \sin ^{-1}\left (\sqrt {\frac {c^2 f}{f c^2+g}}\right ) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g-2 \sqrt {-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b c^2 e f \tanh ^{-1}(c x) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g-2 \sqrt {-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 i b c^2 e f \sin ^{-1}\left (\sqrt {\frac {c^2 f}{f c^2+g}}\right ) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g+2 \sqrt {-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b c^2 e f \tanh ^{-1}(c x) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g+2 \sqrt {-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b e g \tanh ^{-1}(c x) \log \left (\frac {e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2-2 \sqrt {-f} \sqrt {g} c-g}+1\right ) x^2+2 b e g \tanh ^{-1}(c x) \log \left (\frac {e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2+2 \sqrt {-f} \sqrt {g} c-g}+1\right ) x^2-4 a e g \log (x) x^2+2 a e g \log \left (g x^2+f\right ) x^2-2 b c^2 e f \tanh ^{-1}(c x) \log \left (g x^2+f\right ) x^2+2 b c^2 e f \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) x^2+2 b e g \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right ) x^2+b e g \text {Li}_2\left (-\frac {e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2-2 \sqrt {-f} \sqrt {g} c-g}\right ) x^2+b e g \text {Li}_2\left (-\frac {e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2+2 \sqrt {-f} \sqrt {g} c-g}\right ) x^2-b c^2 e f \text {Li}_2\left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (-f c^2+g-2 \sqrt {-c^2 f g}\right )}{f c^2+g}\right ) x^2-b c^2 e f \text {Li}_2\left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (-f c^2+g+2 \sqrt {-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b c d f x+2 b c e f \log \left (g x^2+f\right ) x+2 a d f+2 b d f \tanh ^{-1}(c x)+2 a e f \log \left (g x^2+f\right )+2 b e f \tanh ^{-1}(c x) \log \left (g x^2+f\right )}{4 f x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {artanh}\left (c x\right ) + a d + {\left (b e \operatorname {artanh}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.59, size = 960, normalized size = 2.04 \[ -\frac {d b \,c^{2} \ln \left (c x \right )}{4}+\frac {d b \,c^{2} \ln \left (c x +1\right )}{4}-\frac {d b \ln \left (c x +1\right )}{4 x^{2}}+\frac {d b \,c^{2} \ln \left (-c x \right )}{4}+\frac {d b \ln \left (-c x +1\right )}{4 x^{2}}+\left (-\frac {b e \ln \left (c x +1\right )}{4 x^{2}}+\frac {e \left (b \,c^{2} \ln \left (c x +1\right ) x^{2}-b \,c^{2} \ln \left (-c x +1\right ) x^{2}-2 x b c +b \ln \left (-c x +1\right )-2 a \right )}{4 x^{2}}\right ) \ln \left (g \,x^{2}+f \right )+\frac {a e g \ln \relax (x )}{f}-\frac {a e g \ln \left (g \,x^{2}+f \right )}{2 f}-\frac {d a}{2 x^{2}}-\frac {b \,c^{2} d \ln \left (-c x +1\right )}{4}-\frac {b c d}{2 x}+\frac {b e \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}-\frac {b e \dilog \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \dilog \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \dilog \left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}+\frac {b e \dilog \left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}+\frac {g b e \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \dilog \left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}+\frac {g b e \dilog \left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}-\frac {g b e \dilog \left (c x +1\right )}{2 f}+\frac {g b e \dilog \left (-c x +1\right )}{2 f}-\frac {g b e \dilog \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \dilog \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}-\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}-\frac {g b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g e b c \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{\sqrt {f g}}+\frac {g b e \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f} \]
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 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b d - \frac {1}{2} \, {\left (g {\left (\frac {\log \left (g x^{2} + f\right )}{f} - \frac {\log \left (x^{2}\right )}{f}\right )} + \frac {\log \left (g x^{2} + f\right )}{x^{2}}\right )} a e - \frac {1}{4} \, {\left (2 \, c^{2} g \int \frac {x^{2} \log \left (c x + 1\right )}{g x^{3} + f x}\,{d x} - 2 \, c^{2} g \int \frac {x^{2} \log \left (-c x + 1\right )}{g x^{3} + f x}\,{d x} + \frac {2 i \, c g {\left (\log \left (\frac {i \, g x}{\sqrt {f g}} + 1\right ) - \log \left (-\frac {i \, g x}{\sqrt {f g}} + 1\right )\right )}}{\sqrt {f g}} - 2 \, g \int \frac {\log \left (c x + 1\right )}{g x^{3} + f x}\,{d x} + 2 \, g \int \frac {\log \left (-c x + 1\right )}{g x^{3} + f x}\,{d x} + \frac {{\left (2 \, c x - {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \log \left (g x^{2} + f\right )}{x^{2}}\right )} b e - \frac {a d}{2 \, x^{2}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^3} \,d x \]
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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