Optimal. Leaf size=546 \[ -2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac {b e \text {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-\frac {i b e \sqrt {f} \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {f} \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}} \]
[Out]
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Rubi [A]
time = 0.92, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps
used = 38, number of rules used = 20, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.952, Rules used = {6221, 2525,
2441, 2440, 2438, 6128, 6022, 266, 6122, 211, 6120, 2520, 12, 6820, 4996, 4940, 4966, 2449, 2352,
2497} \begin {gather*} \frac {2 a e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-2 a e x-\frac {b e \sqrt {f} \log \left (1-\frac {1}{c x}\right ) \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \log \left (\frac {1}{c x}+1\right ) \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (-\sqrt {g}+i c \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (c x+1)}{\left (\sqrt {g}+i c \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}+\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 {i b e \sqrt {f} \text {Li}_2\left (\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {f} \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} (c x+1)}{\left (i \sqrt {f} c+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}}-2 b e x \coth ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2449
Rule 2497
Rule 2520
Rule 2525
Rule 4940
Rule 4966
Rule 4996
Rule 6022
Rule 6120
Rule 6122
Rule 6128
Rule 6221
Rule 6820
Rubi steps
\begin {align*} \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx &=x \left (a+b \coth ^{-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 \coth ^{-1}(c x)\right )}{f+g x^2} \, dx\\ &=x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} (b c) \text {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 \coth ^{-1}(c x)\right ) \, dx+(2 e f) \int \frac {a+b \coth ^{-1}(c x)}{f+g x^2} \, dx\\ &=-2 a e x+x \left (a+b \coth ^{-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 \coth ^{-1}(c x) \, dx+(2 a e f) \int \frac {1}{f+g x^2} \, dx+(2 b e f) \int \frac {\coth ^{-1}(c x)}{f+g x^2} \, dx-\frac {(b e g) \text {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-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+x \left (a+b \coth ^{-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 {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 \left (1-\frac {1}{c x}\right )}{f+g x^2} \, dx+(b e f) \int \frac {\log \left (1+\frac {1}{c x}\right )}{f+g x^2} \, dx\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 {(b e f) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (1-\frac {1}{c x}\right ) x^2} \, dx}{c}+\frac {(b e f) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (1+\frac {1}{c x}\right ) x^2} \, dx}{c}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (1-\frac {1}{c x}\right ) x^2} \, dx}{c \sqrt {g}}+\frac {\left (b e \sqrt {f}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (1+\frac {1}{c x}\right ) x^2} \, dx}{c \sqrt {g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 ) \int \frac {c \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (-1+c x)} \, dx}{c \sqrt {g}}+\frac {\left (b e \sqrt {f}\right ) \int \frac {c \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (1+c x)} \, dx}{c \sqrt {g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (-1+c x)} \, dx}{\sqrt {g}}+\frac {\left (b e \sqrt {f}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (1+c x)} \, dx}{\sqrt {g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 ) \int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x}+\frac {c \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{-1+c x}\right ) \, dx}{\sqrt {g}}+\frac {\left (b e \sqrt {f}\right ) \int \left (\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x}-\frac {c \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{1+c x}\right ) \, dx}{\sqrt {g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{-1+c x} \, dx}{\sqrt {g}}-\frac {\left (b c e \sqrt {f}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{1+c x} \, dx}{\sqrt {g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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) \int \frac {\log \left (\frac {2 \sqrt {g} (-1+c x)}{\sqrt {f} \left (i c-\frac {\sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx+(b e) \int \frac {\log \left (\frac {2 \sqrt {g} (1+c x)}{\sqrt {f} \left (i c+\frac {\sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-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 {i b e \sqrt {f} \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {f} \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1287\) vs. \(2(546)=1092\).
time = 2.44, size = 1287, normalized size = 2.36 \begin {gather*} a d x-2 a e x+b d x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {b d \log \left (1-c^2 x^2\right )}{2 c}+a e x \log \left (f+g x^2\right )+b e \left (x \coth ^{-1}(c x)+\frac {\log \left (1-c^2 x^2\right )}{2 c}\right ) \log \left (f+g x^2\right )+\frac {b e \left (-4 c x \coth ^{-1}(c x)+4 \log \left (\frac {1}{c \sqrt {1-\frac {1}{c^2 x^2}} x}\right )+\frac {\sqrt {c^2 f g} \left (-2 i \text {ArcCos}\left (\frac {c^2 f-g}{c^2 f+g}\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 f g}}{c g x}\right )+4 \coth ^{-1}(c x) \text {ArcTan}\left (\frac {c g x}{\sqrt {c^2 f g}}\right )-\left (\text {ArcCos}\left (\frac {c^2 f-g}{c^2 f+g}\right )+2 \text {ArcTan}\left (\frac {\sqrt {c^2 f g}}{c g x}\right )\right ) \log \left (\frac {2 i g \left (i c^2 f+\sqrt {c^2 f g}\right ) \left (-1+\frac {1}{c x}\right )}{\left (c^2 f+g\right ) \left (g+\frac {i \sqrt {c^2 f g}}{c x}\right )}\right )-\left (\text {ArcCos}\left (\frac {c^2 f-g}{c^2 f+g}\right )-2 \text {ArcTan}\left (\frac {\sqrt {c^2 f g}}{c g x}\right )\right ) \log \left (\frac {2 g \left (c^2 f+i \sqrt {c^2 f g}\right ) \left (1+\frac {1}{c x}\right )}{\left (c^2 f+g\right ) \left (g+\frac {i \sqrt {c^2 f g}}{c x}\right )}\right )+\left (\text {ArcCos}\left (\frac {c^2 f-g}{c^2 f+g}\right )+2 \left (\text {ArcTan}\left (\frac {\sqrt {c^2 f g}}{c g x}\right )+\text {ArcTan}\left (\frac {c g x}{\sqrt {c^2 f g}}\right )\right )\right ) \log \left (\frac {\sqrt {2} e^{-\coth ^{-1}(c x)} \sqrt {c^2 f g}}{\sqrt {c^2 f+g} \sqrt {-c^2 f+g+\left (c^2 f+g\right ) \cosh \left (2 \coth ^{-1}(c x)\right )}}\right )+\left (\text {ArcCos}\left (\frac {c^2 f-g}{c^2 f+g}\right )-2 \left (\text {ArcTan}\left (\frac {\sqrt {c^2 f g}}{c g x}\right )+\text {ArcTan}\left (\frac {c g x}{\sqrt {c^2 f g}}\right )\right )\right ) \log \left (\frac {\sqrt {2} e^{\coth ^{-1}(c x)} \sqrt {c^2 f g}}{\sqrt {c^2 f+g} \sqrt {-c^2 f+g+\left (c^2 f+g\right ) \cosh \left (2 \coth ^{-1}(c x)\right )}}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (-c^2 f+g+2 i \sqrt {c^2 f g}\right ) \left (g-\frac {i \sqrt {c^2 f g}}{c x}\right )}{\left (c^2 f+g\right ) \left (g+\frac {i \sqrt {c^2 f g}}{c x}\right )}\right )+\text {PolyLog}\left (2,\frac {\left (c^2 f-g+2 i \sqrt {c^2 f g}\right ) \left (i g+\frac {\sqrt {c^2 f g}}{c x}\right )}{\left (c^2 f+g\right ) \left (-i g+\frac {\sqrt {c^2 f g}}{c x}\right )}\right )\right )\right )}{g}\right )}{2 c}-\frac {b e g \left (\frac {\left (-\log \left (-\frac {1}{c}+x\right )-\log \left (\frac {1}{c}+x\right )+\log \left (1-c^2 x^2\right )\right ) \log \left (f+g x^2\right )}{2 g}+\frac {\log \left (-\frac {1}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (-\frac {1}{c}+x\right )}{-i \sqrt {f}-\frac {\sqrt {g}}{c}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {g} \left (-\frac {1}{c}+x\right )}{-i \sqrt {f}-\frac {\sqrt {g}}{c}}\right )}{2 g}+\frac {\log \left (-\frac {1}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (-\frac {1}{c}+x\right )}{i \sqrt {f}-\frac {\sqrt {g}}{c}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {g} \left (-\frac {1}{c}+x\right )}{i \sqrt {f}-\frac {\sqrt {g}}{c}}\right )}{2 g}+\frac {\log \left (\frac {1}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (\frac {1}{c}+x\right )}{-i \sqrt {f}+\frac {\sqrt {g}}{c}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {g} \left (\frac {1}{c}+x\right )}{-i \sqrt {f}+\frac {\sqrt {g}}{c}}\right )}{2 g}+\frac {\log \left (\frac {1}{c}+x\right ) \log \left (1-\frac {\sqrt {g} \left (\frac {1}{c}+x\right )}{i \sqrt {f}+\frac {\sqrt {g}}{c}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {g} \left (\frac {1}{c}+x\right )}{i \sqrt {f}+\frac {\sqrt {g}}{c}}\right )}{2 g}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 4.22, size = 3508, normalized size = 6.42
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3508\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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