Optimal. Leaf size=101 \[ a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} b d \text {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \text {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} a e \text {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+b e \text {Int}\left (\frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x},x\right ) \]
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Rubi [A]
time = 0.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx &=d \int \frac {a+b \coth ^{-1}(c x)}{x} \, dx+e \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac {1}{2} b d \text {Li}_2\left (-\frac {1}{c x}\right )-\frac {1}{2} b d \text {Li}_2\left (\frac {1}{c x}\right )+(a e) \int \frac {\log \left (f+g x^2\right )}{x} \, dx+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac {1}{2} b d \text {Li}_2\left (-\frac {1}{c x}\right )-\frac {1}{2} b d \text {Li}_2\left (\frac {1}{c x}\right )+\frac {1}{2} (a e) \text {Subst}\left (\int \frac {\log (f+g x)}{x} \, dx,x,x^2\right )+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} b d \text {Li}_2\left (-\frac {1}{c x}\right )-\frac {1}{2} b d \text {Li}_2\left (\frac {1}{c x}\right )+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx-\frac {1}{2} (a e g) \text {Subst}\left (\int \frac {\log \left (-\frac {g x}{f}\right )}{f+g x} \, dx,x,x^2\right )\\ &=a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} b d \text {Li}_2\left (-\frac {1}{c x}\right )-\frac {1}{2} b d \text {Li}_2\left (\frac {1}{c x}\right )+\frac {1}{2} a e \text {Li}_2\left (1+\frac {g x^2}{f}\right )+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (f + g x^{2} \right )}\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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