Optimal. Leaf size=656 \[ x \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 a e x-\frac {b \log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac {b e \text {Li}_2\left (\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )}{2 c}+\frac {b e \log \left (c^2 x^2+1\right )}{c}-\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (i-c x)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (1-i c x)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (i c x+1)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (c x+i)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}-2 b e x \tan ^{-1}(c x) \]
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Rubi [A] time = 0.83, antiderivative size = 656, 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 = {5009, 2475, 2394, 2393, 2391, 4916, 4846, 260, 4910, 205, 4908, 2409} \[ -\frac {b e \text {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f-g}\right )}{2 c}-\frac {i b e \sqrt {-f} \text {PolyLog}\left (2,\frac {\sqrt {g} (-c x+i)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \text {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{\sqrt {g}+i c \sqrt {-f}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \text {PolyLog}\left (2,\frac {\sqrt {g} (1+i c x)}{\sqrt {g}+i c \sqrt {-f}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \text {PolyLog}\left (2,\frac {\sqrt {g} (c x+i)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+x \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 a e x-\frac {b \log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac {b e \log \left (c^2 x^2+1\right )}{c}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}-2 b e x \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2475
Rule 4846
Rule 4908
Rule 4910
Rule 4916
Rule 5009
Rubi steps
\begin {align*} \int \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx &=x \left (a+b \tan ^{-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 \tan ^{-1}(c x)\right )}{f+g x^2} \, dx\\ &=x \left (a+b \tan ^{-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 \tan ^{-1}(c x)\right ) \, dx+(2 e f) \int \frac {a+b \tan ^{-1}(c x)}{f+g x^2} \, dx\\ &=-2 a e x+x \left (a+b \tan ^{-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 \tan ^{-1}(c x) \, dx+(2 a e f) \int \frac {1}{f+g x^2} \, dx+(2 b e f) \int \frac {\tan ^{-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-2 b e x \tan ^{-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 \tan ^{-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+(i b e f) \int \frac {\log (1-i c x)}{f+g x^2} \, dx-(i b e f) \int \frac {\log (1+i c x)}{f+g x^2} \, dx\\ &=-2 a e x-2 b e x \tan ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {b e \log \left (1+c^2 x^2\right )}{c}+x \left (a+b \tan ^{-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}+(i b e f) \int \left (\frac {\sqrt {-f} \log (1-i c x)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (1-i c x)}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx-(i b e f) \int \left (\frac {\sqrt {-f} \log (1+i c x)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (1+i c x)}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx\\ &=-2 a e x-2 b e x \tan ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {b e \log \left (1+c^2 x^2\right )}{c}+x \left (a+b \tan ^{-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 (i b e \sqrt {-f}\right ) \int \frac {\log (1-i c x)}{\sqrt {-f}-\sqrt {g} x} \, dx+\frac {1}{2} \left (i b e \sqrt {-f}\right ) \int \frac {\log (1-i c x)}{\sqrt {-f}+\sqrt {g} x} \, dx-\frac {1}{2} \left (i b e \sqrt {-f}\right ) \int \frac {\log (1+i c x)}{\sqrt {-f}-\sqrt {g} x} \, dx-\frac {1}{2} \left (i b e \sqrt {-f}\right ) \int \frac {\log (1+i c x)}{\sqrt {-f}+\sqrt {g} x} \, dx\\ &=-2 a e x-2 b e x \tan ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \log \left (1+c^2 x^2\right )}{c}+x \left (a+b \tan ^{-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 {i c \left (\sqrt {-f}-\sqrt {g} x\right )}{-i c \sqrt {-f}+\sqrt {g}}\right )}{1-i c x} \, dx}{2 \sqrt {g}}+\frac {\left (b c e \sqrt {-f}\right ) \int \frac {\log \left (\frac {i c \left (\sqrt {-f}-\sqrt {g} x\right )}{i c \sqrt {-f}+\sqrt {g}}\right )}{1+i c x} \, dx}{2 \sqrt {g}}-\frac {\left (b c e \sqrt {-f}\right ) \int \frac {\log \left (-\frac {i c \left (\sqrt {-f}+\sqrt {g} x\right )}{-i c \sqrt {-f}-\sqrt {g}}\right )}{1-i c x} \, dx}{2 \sqrt {g}}-\frac {\left (b c e \sqrt {-f}\right ) \int \frac {\log \left (\frac {i c \left (\sqrt {-f}+\sqrt {g} x\right )}{i c \sqrt {-f}-\sqrt {g}}\right )}{1+i c x} \, dx}{2 \sqrt {g}}\\ &=-2 a e x-2 b e x \tan ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \log \left (1+c^2 x^2\right )}{c}+x \left (a+b \tan ^{-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 (i b e \sqrt {-f}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{-i c \sqrt {-f}-\sqrt {g}}\right )}{x} \, dx,x,1-i c x\right )}{2 \sqrt {g}}+\frac {\left (i b e \sqrt {-f}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{i c \sqrt {-f}-\sqrt {g}}\right )}{x} \, dx,x,1+i c x\right )}{2 \sqrt {g}}+\frac {\left (i b e \sqrt {-f}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{-i c \sqrt {-f}+\sqrt {g}}\right )}{x} \, dx,x,1-i c x\right )}{2 \sqrt {g}}-\frac {\left (i b e \sqrt {-f}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{i c \sqrt {-f}+\sqrt {g}}\right )}{x} \, dx,x,1+i c x\right )}{2 \sqrt {g}}\\ &=-2 a e x-2 b e x \tan ^{-1}(c x)+\frac {2 a e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \log \left (1+c^2 x^2\right )}{c}+x \left (a+b \tan ^{-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 {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (i-c x)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (1-i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (1+i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {i b e \sqrt {-f} \text {Li}_2\left (\frac {\sqrt {g} (i+c x)}{c \sqrt {-f}+i \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 [B] time = 4.55, size = 1352, normalized size = 2.06 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b d \arctan \left (c x\right ) + a d + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right ), x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.99, size = 0, normalized size = 0.00 \[ \int \left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )\, dx \]
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 \[ {\left (2 \, g {\left (\frac {f \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} g} - \frac {x}{g}\right )} + x \log \left (g x^{2} + f\right )\right )} a e + a d x + b e \int \arctan \left (c x\right ) \log \left (g x^{2} + f\right )\,{d x} + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \]
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 \[ \int \left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]
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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