Optimal. Leaf size=780 \[ -\frac {\text {Li}_2\left (\frac {\sqrt [3]{d} (-a-b x+1)}{\sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \text {Li}_2\left (-\frac {\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+1)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{d} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\text {Li}_2\left (-\frac {\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{d} (a+1)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
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Rubi [A] time = 1.40, antiderivative size = 780, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6115, 2409, 2394, 2393, 2391} \[ -\frac {\text {PolyLog}\left (2,\frac {\sqrt [3]{d} (-a-b x+1)}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \text {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+1)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \text {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{d} (-a-b x+1)}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\text {PolyLog}\left (2,-\frac {\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \text {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 6115
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{c+d x^3} \, dx &=-\left (\frac {1}{2} \int \frac {\log (1-a-b x)}{c+d x^3} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+d x^3} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {\log (1-a-b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}-\frac {\log (1-a-b x)}{3 c^{2/3} \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}-\frac {\log (1-a-b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}\right ) \, dx\right )+\frac {1}{2} \int \left (-\frac {\log (1+a+b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}-\frac {\log (1+a+b x)}{3 c^{2/3} \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}-\frac {\log (1+a+b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (1-a-b x)}{-\sqrt [3]{c}-\sqrt [3]{d} x} \, dx}{6 c^{2/3}}+\frac {\int \frac {\log (1-a-b x)}{-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}+\frac {\int \frac {\log (1-a-b x)}{-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}-\frac {\int \frac {\log (1+a+b x)}{-\sqrt [3]{c}-\sqrt [3]{d} x} \, dx}{6 c^{2/3}}-\frac {\int \frac {\log (1+a+b x)}{-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}-\frac {\int \frac {\log (1+a+b x)}{-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}\\ &=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {b \int \frac {\log \left (-\frac {b \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}-\frac {b \int \frac {\log \left (\frac {b \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}{-b \sqrt [3]{c}+(1+a) \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}+\frac {\left (\sqrt [3]{-1} b\right ) \int \frac {\log \left (-\frac {b \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}+\frac {\left (\sqrt [3]{-1} b\right ) \int \frac {\log \left (\frac {b \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}{-b \sqrt [3]{c}+(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}-\frac {\left ((-1)^{2/3} b\right ) \int \frac {\log \left (-\frac {b \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}-\frac {\left ((-1)^{2/3} b\right ) \int \frac {\log \left (\frac {b \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}{-b \sqrt [3]{c}-\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}\\ &=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{d} x}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{d} x}{-b \sqrt [3]{c}+(1+a) \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {(-1)^{2/3} \sqrt [3]{d} x}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {(-1)^{2/3} \sqrt [3]{d} x}{-b \sqrt [3]{c}+(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{-1} \sqrt [3]{d} x}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{-1} \sqrt [3]{d} x}{-b \sqrt [3]{c}-\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{2/3} \sqrt [3]{d}}\\ &=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\text {Li}_2\left (\frac {\sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \text {Li}_2\left (-\frac {\sqrt [3]{-1} \sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\text {Li}_2\left (-\frac {\sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 623, normalized size = 0.80 \[ \frac {-\text {Li}_2\left (-\frac {\sqrt [3]{d} (a+b x-1)}{b \sqrt [3]{c}-(a-1) \sqrt [3]{d}}\right )-(-1)^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{d} (a+b x-1)}{\sqrt [3]{-1} \sqrt [3]{d} (a-1)+b \sqrt [3]{c}}\right )+\sqrt [3]{-1} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{d} (a+b x-1)}{(-1)^{2/3} (a-1) \sqrt [3]{d}-b \sqrt [3]{c}}\right )+\text {Li}_2\left (-\frac {\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )+(-1)^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{d} (a+1)+b \sqrt [3]{c}}\right )-\sqrt [3]{-1} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{d} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{d}-b \sqrt [3]{c}}\right )-\log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a-1) \sqrt [3]{d}}\right )+\log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )-(-1)^{2/3} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a-1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )+(-1)^{2/3} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )+\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a-1) \sqrt [3]{d}}\right )-\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (b x + a\right )}{d x^{3} + c}, x\right ) \]
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 [C] time = 0.91, size = 587, normalized size = 0.75 \[ -\frac {2 b^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (\left (a^{3} d -c \,b^{3}-3 a^{2} d +3 a d -d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d -3 c \,b^{3}-3 a^{2} d -3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 c \,b^{3}+3 a^{2} d -3 a d -3 d \right ) \textit {\_Z}^{2}+a^{3} d -c \,b^{3}+3 a^{2} d +3 a d +d \right )}{\sum }\frac {\arctanh \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} a^{3} d -\textit {\_R1}^{4} b^{3} c -3 \textit {\_R1}^{4} a^{2} d +3 \textit {\_R1}^{4} a d +2 \textit {\_R1}^{2} a^{3} d -2 \textit {\_R1}^{2} b^{3} c -\textit {\_R1}^{4} d -2 \textit {\_R1}^{2} a^{2} d -2 \textit {\_R1}^{2} a d +a^{3} d -c \,b^{3}+2 \textit {\_R1}^{2} d +a^{2} d -a d -d}\right )}{3}-\frac {2 b^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (\left (a^{3} d -c \,b^{3}-3 a^{2} d +3 a d -d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d -3 c \,b^{3}-3 a^{2} d -3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 c \,b^{3}+3 a^{2} d -3 a d -3 d \right ) \textit {\_Z}^{2}+a^{3} d -c \,b^{3}+3 a^{2} d +3 a d +d \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\arctanh \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} a^{3} d -\textit {\_R1}^{4} b^{3} c -3 \textit {\_R1}^{4} a^{2} d +3 \textit {\_R1}^{4} a d +2 \textit {\_R1}^{2} a^{3} d -2 \textit {\_R1}^{2} b^{3} c -\textit {\_R1}^{4} d -2 \textit {\_R1}^{2} a^{2} d -2 \textit {\_R1}^{2} a d +a^{3} d -c \,b^{3}+2 \textit {\_R1}^{2} d +a^{2} d -a d -d}\right )}{3} \]
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 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{d x^{3} + c}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{d\,x^3+c} \,d x \]
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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