Optimal. Leaf size=613 \[ -\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \text {Li}_2\left (\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} b c e \text {Li}_2\left (\frac {g x^2}{f}+1\right )-\frac {b e \sqrt {g} \text {Li}_2\left (-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \text {Li}_2\left (\frac {\sqrt {g} (1-c x)}{\sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \text {Li}_2\left (-\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \text {Li}_2\left (\frac {\sqrt {g} (c x+1)}{\sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}} \]
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Rubi [A] time = 0.74, antiderivative size = 613, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6081, 2475, 36, 29, 31, 2416, 2394, 2315, 2393, 2391, 5974, 205, 5972, 2409} \[ -\frac {1}{2} b c e \text {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \text {PolyLog}\left (2,\frac {g x^2}{f}+1\right )-\frac {b e \sqrt {g} \text {PolyLog}\left (2,-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \text {PolyLog}\left (2,\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \text {PolyLog}\left (2,-\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \text {PolyLog}\left (2,\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 205
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2416
Rule 2475
Rule 5972
Rule 5974
Rule 6081
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^2} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+(b c) \int \frac {d+e \log \left (f+g x^2\right )}{x \left (1-c^2 x^2\right )} \, dx+(2 e g) \int \frac {a+b \tanh ^{-1}(c x)}{f+g x^2} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (f+g x)}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+(2 a e g) \int \frac {1}{f+g x^2} \, dx+(2 b e g) \int \frac {\tanh ^{-1}(c x)}{f+g x^2} \, dx\\ &=\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \left (\frac {d+e \log (f+g x)}{x}-\frac {c^2 (d+e \log (f+g x))}{-1+c^2 x}\right ) \, dx,x,x^2\right )-(b e g) \int \frac {\log (1-c x)}{f+g x^2} \, dx+(b e g) \int \frac {\log (1+c x)}{f+g x^2} \, dx\\ &=\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (f+g x)}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {d+e \log (f+g x)}{-1+c^2 x} \, dx,x,x^2\right )-(b e g) \int \left (\frac {\sqrt {-f} \log (1-c x)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (1-c x)}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx+(b e g) \int \left (\frac {\sqrt {-f} \log (1+c x)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (1+c x)}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx\\ &=\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} (b c e g) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {g x}{f}\right )}{f+g x} \, dx,x,x^2\right )+\frac {1}{2} (b c e g) \operatorname {Subst}\left (\int \frac {\log \left (\frac {g \left (-1+c^2 x\right )}{-c^2 f-g}\right )}{f+g x} \, dx,x,x^2\right )+\frac {(b e g) \int \frac {\log (1-c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {-f}}+\frac {(b e g) \int \frac {\log (1-c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {-f}}-\frac {(b e g) \int \frac {\log (1+c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {-f}}-\frac {(b e g) \int \frac {\log (1+c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {-f}}\\ &=\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} b c e \text {Li}_2\left (1+\frac {g x^2}{f}\right )+\frac {1}{2} (b c e) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {c^2 x}{-c^2 f-g}\right )}{x} \, dx,x,f+g x^2\right )-\frac {\left (b c e \sqrt {g}\right ) \int \frac {\log \left (-\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{-c \sqrt {-f}+\sqrt {g}}\right )}{1-c x} \, dx}{2 \sqrt {-f}}-\frac {\left (b c e \sqrt {g}\right ) \int \frac {\log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{1+c x} \, dx}{2 \sqrt {-f}}+\frac {\left (b c e \sqrt {g}\right ) \int \frac {\log \left (-\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{-c \sqrt {-f}-\sqrt {g}}\right )}{1-c x} \, dx}{2 \sqrt {-f}}+\frac {\left (b c e \sqrt {g}\right ) \int \frac {\log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{1+c x} \, dx}{2 \sqrt {-f}}\\ &=\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \text {Li}_2\left (1+\frac {g x^2}{f}\right )-\frac {\left (b e \sqrt {g}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{-c \sqrt {-f}-\sqrt {g}}\right )}{x} \, dx,x,1-c x\right )}{2 \sqrt {-f}}+\frac {\left (b e \sqrt {g}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{c \sqrt {-f}-\sqrt {g}}\right )}{x} \, dx,x,1+c x\right )}{2 \sqrt {-f}}+\frac {\left (b e \sqrt {g}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{-c \sqrt {-f}+\sqrt {g}}\right )}{x} \, dx,x,1-c x\right )}{2 \sqrt {-f}}-\frac {\left (b e \sqrt {g}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{c \sqrt {-f}+\sqrt {g}}\right )}{x} \, dx,x,1+c x\right )}{2 \sqrt {-f}}\\ &=\frac {2 a e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b e \sqrt {g} \text {Li}_2\left (-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \text {Li}_2\left (\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \text {Li}_2\left (-\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \text {Li}_2\left (\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {1}{2} b c e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \text {Li}_2\left (1+\frac {g x^2}{f}\right )\\ \end {align*}
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Mathematica [C] time = 3.27, size = 1226, normalized size = 2.00 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {artanh}\left (c x\right ) + a d + {\left (b e \operatorname {artanh}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctanh \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x^{2}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b d + {\left (\frac {2 \, g \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}} - \frac {\log \left (g x^{2} + f\right )}{x}\right )} a e + \frac {1}{2} \, b e \int \frac {{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \log \left (g x^{2} + f\right )}{x^{2}}\,{d x} - \frac {a d}{x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^2} \,d x \]
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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