Optimal. Leaf size=832 \[ \frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {\sqrt [3]{d} \log \left (\frac {b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \log \left (-\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}-\frac {\sqrt [3]{d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (a+b x+1) \log \left (-\frac {b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{c} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{c} (a+b x+1)}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{c} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{c} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}} \]
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Rubi [A] time = 1.43, antiderivative size = 832, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6115, 2409, 2389, 2295, 2394, 2393, 2391} \[ \frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {\sqrt [3]{d} \log \left (\frac {b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \log \left (-\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}-\frac {\sqrt [3]{d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (a+b x+1) \log \left (-\frac {b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \text {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{d} \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (a+b x+1)}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \text {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 6115
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a+b x)}{c+\frac {d}{x^3}} \, dx &=-\left (\frac {1}{2} \int \frac {\log (1-a-b x)}{c+\frac {d}{x^3}} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+\frac {d}{x^3}} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\log (1-a-b x)}{c}-\frac {d \log (1-a-b x)}{c \left (d+c x^3\right )}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\log (1+a+b x)}{c}-\frac {d \log (1+a+b x)}{c \left (d+c x^3\right )}\right ) \, dx\\ &=-\frac {\int \log (1-a-b x) \, dx}{2 c}+\frac {\int \log (1+a+b x) \, dx}{2 c}+\frac {d \int \frac {\log (1-a-b x)}{d+c x^3} \, dx}{2 c}-\frac {d \int \frac {\log (1+a+b x)}{d+c x^3} \, dx}{2 c}\\ &=\frac {\operatorname {Subst}(\int \log (x) \, dx,x,1-a-b x)}{2 b c}+\frac {\operatorname {Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}+\frac {d \int \left (-\frac {\log (1-a-b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac {\log (1-a-b x)}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac {\log (1-a-b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{2 c}-\frac {d \int \left (-\frac {\log (1+a+b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac {\log (1+a+b x)}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac {\log (1+a+b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{2 c}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}-\frac {\sqrt [3]{d} \int \frac {\log (1-a-b x)}{-\sqrt [3]{d}-\sqrt [3]{c} x} \, dx}{6 c}-\frac {\sqrt [3]{d} \int \frac {\log (1-a-b x)}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x} \, dx}{6 c}-\frac {\sqrt [3]{d} \int \frac {\log (1-a-b x)}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x} \, dx}{6 c}+\frac {\sqrt [3]{d} \int \frac {\log (1+a+b x)}{-\sqrt [3]{d}-\sqrt [3]{c} x} \, dx}{6 c}+\frac {\sqrt [3]{d} \int \frac {\log (1+a+b x)}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x} \, dx}{6 c}+\frac {\sqrt [3]{d} \int \frac {\log (1+a+b x)}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x} \, dx}{6 c}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}-\frac {\sqrt [3]{d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\left (b \sqrt [3]{d}\right ) \int \frac {\log \left (\frac {b \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{4/3}}+\frac {\left (b \sqrt [3]{d}\right ) \int \frac {\log \left (-\frac {b \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{4/3}}-\frac {\left (\sqrt [3]{-1} b \sqrt [3]{d}\right ) \int \frac {\log \left (\frac {b \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{4/3}}-\frac {\left (\sqrt [3]{-1} b \sqrt [3]{d}\right ) \int \frac {\log \left (-\frac {b \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{4/3}}+\frac {\left ((-1)^{2/3} b \sqrt [3]{d}\right ) \int \frac {\log \left (\frac {b \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{-\sqrt [3]{-1} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{4/3}}+\frac {\left ((-1)^{2/3} b \sqrt [3]{d}\right ) \int \frac {\log \left (-\frac {b \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{-\sqrt [3]{-1} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{4/3}}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}-\frac {\sqrt [3]{d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{c} x}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{4/3}}-\frac {\sqrt [3]{d} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{c} x}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{4/3}}-\frac {\left (\sqrt [3]{-1} \sqrt [3]{d}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {(-1)^{2/3} \sqrt [3]{c} x}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{4/3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{d}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {(-1)^{2/3} \sqrt [3]{c} x}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{4/3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{d}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{-\sqrt [3]{-1} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{4/3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{d}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{-\sqrt [3]{-1} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{4/3}}\\ &=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}-\frac {\sqrt [3]{d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{c} (1-a-b x)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{c} (1-a-b x)}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{c} (1-a-b x)}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {\sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{c} (1+a+b x)}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{c} (1+a+b x)}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{c} (1+a+b x)}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}\\ \end {align*}
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Mathematica [C] time = 8.06, size = 917, normalized size = 1.10 \[ -\frac {d \text {RootSum}\left [c \text {$\#$1}^3 a^3+3 c \text {$\#$1}^2 a^3+c a^3+3 c \text {$\#$1} a^3-3 c \text {$\#$1}^3 a^2-3 c \text {$\#$1}^2 a^2+3 c a^2+3 c \text {$\#$1} a^2+3 c \text {$\#$1}^3 a-3 c \text {$\#$1}^2 a+3 c a-3 c \text {$\#$1} a-c \text {$\#$1}^3-b^3 d \text {$\#$1}^3+3 c \text {$\#$1}^2-3 b^3 d \text {$\#$1}^2+c-b^3 d-3 c \text {$\#$1}-3 b^3 d \text {$\#$1}\& ,\frac {-2 \text {$\#$1} \tanh ^{-1}(a+b x)^2+2 e^{-\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )} \sqrt {\frac {\text {$\#$1}}{(\text {$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2+2 e^{-\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )} \text {$\#$1}^2 \sqrt {\frac {\text {$\#$1}}{(\text {$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2+4 e^{-\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )} \text {$\#$1} \sqrt {\frac {\text {$\#$1}}{(\text {$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2-2 \tanh ^{-1}(a+b x)^2+2 \tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right ) \text {$\#$1}^2 \tanh ^{-1}(a+b x)+2 \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )}\right ) \text {$\#$1}^2 \tanh ^{-1}(a+b x)+i \pi \text {$\#$1}^2 \tanh ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right ) \tanh ^{-1}(a+b x)-2 \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )}\right ) \tanh ^{-1}(a+b x)-i \pi \tanh ^{-1}(a+b x)-i \pi \log \left (1+e^{2 \tanh ^{-1}(a+b x)}\right ) \text {$\#$1}^2+2 \tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )}\right ) \text {$\#$1}^2+i \pi \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right ) \text {$\#$1}^2-2 \tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right ) \log \left (i \sinh \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )\right ) \text {$\#$1}^2-\text {Li}_2\left (e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )}\right ) \text {$\#$1}^2+i \pi \log \left (1+e^{2 \tanh ^{-1}(a+b x)}\right )-2 \tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )}\right )-i \pi \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )+2 \tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right ) \log \left (i \sinh \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )\right )+\text {Li}_2\left (e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{\text {$\#$1}+1}\right )\right )}\right )}{c \text {$\#$1}^2 a^3+c a^3+2 c \text {$\#$1} a^3-2 c \text {$\#$1}^2 a^2+2 c a^2+c \text {$\#$1}^2 a+c a-2 c \text {$\#$1} a-b^3 d \text {$\#$1}^2-b^3 d-2 b^3 d \text {$\#$1}}\& \right ] b^3-6 (a+b x) \tanh ^{-1}(a+b x)+6 \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )}{6 b c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \operatorname {artanh}\left (b x + a\right )}{c x^{3} + d}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.89, size = 650, normalized size = 0.78 \[ \frac {\arctanh \left (b x +a \right ) x}{c}+\frac {\arctanh \left (b x +a \right ) a}{b c}+\frac {2 b^{2} d \left (\munderset {\textit {\_R1} =\RootOf \left (\left (a^{3} c -d \,b^{3}-3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 d \,b^{3}-3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 d \,b^{3}+3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -d \,b^{3}+3 a^{2} c +3 a c +c \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} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -d \,b^{3}+2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3 c}+\frac {2 b^{2} d \left (\munderset {\textit {\_R1} =\RootOf \left (\left (a^{3} c -d \,b^{3}-3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 d \,b^{3}-3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 d \,b^{3}+3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -d \,b^{3}+3 a^{2} c +3 a c +c \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} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -d \,b^{3}+2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3 c}+\frac {\ln \left (b x +a -1\right )}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c} \]
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 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x^{3}}}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{x^3}} \,d x \]
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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