Optimal. Leaf size=523 \[ \frac {\log (d+e x) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (\frac {e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+\sqrt [3]{-1}\right )}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+(-1)^{2/3}\right )}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (\frac {(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e} \]
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Rubi [F] time = 0.06, 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 = {} \[ \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{d+e x} \, dx &=\int \left (\frac {a}{d+e x}+\frac {b \tanh ^{-1}\left (c x^3\right )}{d+e x}\right ) \, dx\\ &=\frac {a \log (d+e x)}{e}+b \int \frac {\tanh ^{-1}\left (c x^3\right )}{d+e x} \, dx\\ \end {align*}
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Mathematica [C] time = 100.90, size = 515, normalized size = 0.98 \[ \frac {a \log (d+e x)}{e}+\frac {b \left (-\text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )+\text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )+\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-i \sqrt {3} e-e}\right )-\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-i \sqrt {3} e+e}\right )+\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i \sqrt {3} e-e}\right )-\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i \sqrt {3} e+e}\right )+2 \tanh ^{-1}\left (c x^3\right ) \log (d+e x)-\log (d+e x) \log \left (\frac {e \left (-2 \sqrt [3]{c} x-i \sqrt {3}+1\right )}{2 \sqrt [3]{c} d-i \sqrt {3} e+e}\right )+\log (d+e x) \log \left (\frac {e \left (-2 i \sqrt [3]{c} x+\sqrt {3}-i\right )}{2 i \sqrt [3]{c} d+\left (\sqrt {3}-i\right ) e}\right )+\log (d+e x) \log \left (\frac {e \left (2 i \sqrt [3]{c} x+\sqrt {3}+i\right )}{\left (\sqrt {3}+i\right ) e-2 i \sqrt [3]{c} d}\right )-\log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )-\log (d+e x) \log \left (-\frac {e \left (2 \sqrt [3]{c} x-i \sqrt {3}-1\right )}{2 \sqrt [3]{c} d+i \sqrt {3} e+e}\right )+\log (d+e x) \log \left (\frac {e-\sqrt [3]{c} e x}{\sqrt [3]{c} d+e}\right )\right )}{2 e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 182, normalized size = 0.35 \[ \frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctanh \left (c \,x^{3}\right )}{e}-\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3} c -3 c d \,\textit {\_Z}^{2}+3 c \,d^{2} \textit {\_Z} -c \,d^{3}+e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3} c -3 c d \,\textit {\_Z}^{2}+3 c \,d^{2} \textit {\_Z} -c \,d^{3}-e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e} \]
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 \[ \frac {1}{2} \, b \int \frac {\log \left (c x^{3} + 1\right ) - \log \left (-c x^{3} + 1\right )}{e x + d}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^3\right )}{d+e\,x} \,d x \]
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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