Optimal. Leaf size=599 \[ x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 a e x+\frac {b \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac {b e \text {Li}_2\left (\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{2 c}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+\frac {b e \sqrt {-f} \text {Li}_2\left (-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (1-c x)}{\sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \text {Li}_2\left (-\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (c x+1)}{\sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-2 b e x \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.80, antiderivative size = 599, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6073, 2475, 2394, 2393, 2391, 5980, 5910, 260, 5974, 205, 5972, 2409} \[ \frac {b e \text {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac {b e \sqrt {-f} \text {PolyLog}\left (2,-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \text {PolyLog}\left (2,\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \text {PolyLog}\left (2,-\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \text {PolyLog}\left (2,\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 a e x+\frac {b \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (c x+1) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-2 b e x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2475
Rule 5910
Rule 5972
Rule 5974
Rule 5980
Rule 6073
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx &=x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-(b c) \int \frac {x \left (d+e \log \left (f+g x^2\right )\right )}{1-c^2 x^2} \, dx-(2 e g) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{f+g x^2} \, dx\\ &=x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (f+g x)}{1-c^2 x} \, dx,x,x^2\right )-(2 e) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+(2 e f) \int \frac {a+b \tanh ^{-1}(c x)}{f+g x^2} \, dx\\ &=-2 a e x+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}-(2 b e) \int \tanh ^{-1}(c x) \, dx+(2 a e f) \int \frac {1}{f+g x^2} \, dx+(2 b e f) \int \frac {\tanh ^{-1}(c x)}{f+g x^2} \, dx-\frac {(b 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 )}{2 c}\\ &=-2 a e x+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \tanh ^{-1}(c x)+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}-\frac {(b 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 )}{2 c}+(2 b c e) \int \frac {x}{1-c^2 x^2} \, dx-(b e f) \int \frac {\log (1-c x)}{f+g x^2} \, dx+(b e f) \int \frac {\log (1+c x)}{f+g x^2} \, dx\\ &=-2 a e x+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \tanh ^{-1}(c x)-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}+\frac {b e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-(b e f) \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 f) \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\\ &=-2 a e x+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \tanh ^{-1}(c x)-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}+\frac {b e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-\frac {1}{2} \left (b e \sqrt {-f}\right ) \int \frac {\log (1-c x)}{\sqrt {-f}-\sqrt {g} x} \, dx-\frac {1}{2} \left (b e \sqrt {-f}\right ) \int \frac {\log (1-c x)}{\sqrt {-f}+\sqrt {g} x} \, dx+\frac {1}{2} \left (b e \sqrt {-f}\right ) \int \frac {\log (1+c x)}{\sqrt {-f}-\sqrt {g} x} \, dx+\frac {1}{2} \left (b e \sqrt {-f}\right ) \int \frac {\log (1+c x)}{\sqrt {-f}+\sqrt {g} x} \, dx\\ &=-2 a e x+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \tanh ^{-1}(c x)+\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}+\frac {b e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac {\left (b c e \sqrt {-f}\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 {g}}+\frac {\left (b c e \sqrt {-f}\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 {g}}-\frac {\left (b c e \sqrt {-f}\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 {g}}-\frac {\left (b c e \sqrt {-f}\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 {g}}\\ &=-2 a e x+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \tanh ^{-1}(c x)+\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}+\frac {b e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac {\left (b e \sqrt {-f}\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 {g}}-\frac {\left (b e \sqrt {-f}\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 {g}}-\frac {\left (b e \sqrt {-f}\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 {g}}+\frac {\left (b e \sqrt {-f}\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 {g}}\\ &=-2 a e x+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \tanh ^{-1}(c x)+\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \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 )}{2 c}+\frac {b e \sqrt {-f} \text {Li}_2\left (-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \text {Li}_2\left (-\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \text {Li}_2\left (\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}\\ \end {align*}
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Mathematica [C] time = 3.36, size = 1251, normalized size = 2.09 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b d \operatorname {artanh}\left (c x\right ) + a d + {\left (b e \operatorname {artanh}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right ), x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.63, size = 0, normalized size = 0.00 \[ \int \left (a +b \arctanh \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )\, dx \]
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 \[ {\left (2 \, g {\left (\frac {f \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} g} - \frac {x}{g}\right )} + x \log \left (g x^{2} + f\right )\right )} a e + a d x + \frac {1}{2} \, b e {\left (\frac {{\left ({\left (c x + 1\right )} \log \left (c x + 1\right ) - {\left (c x - 1\right )} \log \left (-c x + 1\right )\right )} \log \left (g x^{2} + f\right )}{c} + \int -\frac {2 \, {\left ({\left (c g x^{2} + g x\right )} \log \left (c x + 1\right ) - {\left (c g x^{2} - g x\right )} \log \left (-c x + 1\right )\right )}}{c g x^{2} + c f}\,{d x}\right )} + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \]
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 \[ \int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]
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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