Optimal. Leaf size=88 \[ \frac {1}{6} c^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2-\frac {b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{3 x^3}-\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 x^6}-\frac {1}{6} b^2 c^2 \log \left (1-c^2 x^6\right )+b^2 c^2 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 1.06, antiderivative size = 360, normalized size of antiderivative = 4.09, number of steps used = 46, number of rules used = 23, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.438, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2395, 44, 2439, 2416, 36, 29, 2392, 2391, 2394, 2393, 2410, 2390} \[ -\frac {1}{12} b^2 c^2 \text {PolyLog}\left (2,\frac {1}{2} \left (1-c x^3\right )\right )-\frac {1}{12} b^2 c^2 \text {PolyLog}\left (2,\frac {1}{2} \left (c x^3+1\right )\right )+\frac {1}{12} b c^2 \log \left (\frac {1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^6}-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (c x^3+1\right )-\frac {1}{12} b^2 c^2 \log \left (1-c x^3\right )-\frac {1}{6} b^2 c^2 \log \left (c x^3+1\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )+b^2 c^2 \log (x)-\frac {b^2 \log ^2\left (c x^3+1\right )}{24 x^6}-\frac {b^2 c \log \left (c x^3+1\right )}{6 x^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 44
Rule 2301
Rule 2314
Rule 2315
Rule 2316
Rule 2344
Rule 2347
Rule 2390
Rule 2391
Rule 2392
Rule 2393
Rule 2394
Rule 2395
Rule 2398
Rule 2410
Rule 2411
Rule 2416
Rule 2439
Rule 2454
Rule 6099
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{x^7} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{4 x^7}-\frac {b \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{2 x^7}+\frac {b^2 \log ^2\left (1+c x^3\right )}{4 x^7}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{x^7} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{x^7} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+c x^3\right )}{x^7} \, dx\\ &=\frac {1}{12} \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^3} \, dx,x,x^3\right )-\frac {1}{6} b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log (1+c x)}{x^3} \, dx,x,x^3\right )+\frac {1}{12} b^2 \operatorname {Subst}\left (\int \frac {\log ^2(1+c x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}+\frac {1}{12} (b c) \operatorname {Subst}\left (\int \frac {2 a-b \log (1-c x)}{x^2 (1-c x)} \, dx,x,x^3\right )-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{x^2 (1+c x)} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^2 (1-c x)} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^2 (1+c x)} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} b \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \left (\frac {-2 a+b \log (1-c x)}{x^2}-\frac {c (-2 a+b \log (1-c x))}{x}+\frac {c^2 (-2 a+b \log (1-c x))}{1+c x}\right ) \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log (1+c x)}{x^2}+\frac {c \log (1+c x)}{x}-\frac {c^2 \log (1+c x)}{-1+c x}\right ) \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log (1+c x)}{x^2}-\frac {c \log (1+c x)}{x}+\frac {c^2 \log (1+c x)}{1+c x}\right ) \, dx,x,x^3\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} b \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-c x^3\right )-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{x^2} \, dx,x,x^3\right )+2 \left (\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,x^3\right )\right )+\frac {1}{12} \left (b c^2\right ) \operatorname {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{x} \, dx,x,x^3\right )-\frac {1}{12} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=-\frac {1}{2} a b c^2 \log (x)-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )-\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )-\frac {1}{12} \left (b c^2\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (1-c x)} \, dx,x,x^3\right )+2 \left (-\frac {b^2 c \log \left (1+c x^3\right )}{12 x^3}+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+c x)} \, dx,x,x^3\right )\right )+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )\\ &=\frac {1}{4} b^2 c^2 \log (x)-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}+\frac {1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac {1}{12} b^2 c^2 \text {Li}_2\left (c x^3\right )+\frac {1}{12} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x} \, dx,x,x^3\right )+2 \left (-\frac {b^2 c \log \left (1+c x^3\right )}{12 x^3}+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c x} \, dx,x,x^3\right )\right )\\ &=\frac {1}{2} b^2 c^2 \log (x)-\frac {1}{12} b^2 c^2 \log \left (1-c x^3\right )-\frac {b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}+\frac {1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{24 x^6}+2 \left (\frac {1}{4} b^2 c^2 \log (x)-\frac {1}{12} b^2 c^2 \log \left (1+c x^3\right )-\frac {b^2 c \log \left (1+c x^3\right )}{12 x^3}\right )-\frac {1}{12} b^2 c^2 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )-\frac {1}{12} b^2 c^2 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 111, normalized size = 1.26 \[ \frac {1}{6} \left (-\frac {a^2}{x^6}-b c^2 (a+b) \log \left (1-c x^3\right )+b c^2 (a-b) \log \left (c x^3+1\right )-\frac {2 a b c}{x^3}-\frac {2 b \tanh ^{-1}\left (c x^3\right ) \left (a+b c x^3\right )}{x^6}+\frac {b^2 \left (c^2 x^6-1\right ) \tanh ^{-1}\left (c x^3\right )^2}{x^6}+6 b^2 c^2 \log (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.36, size = 151, normalized size = 1.72 \[ \frac {24 \, b^{2} c^{2} x^{6} \log \relax (x) + 4 \, {\left (a b - b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} - 1\right ) - 8 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c x^{3} + a b\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{24 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{2}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.28, size = 257, normalized size = 2.92 \[ \frac {b^{2} \left (c^{2} x^{6}-1\right ) \ln \left (c \,x^{3}+1\right )^{2}}{24 x^{6}}-\frac {b \left (x^{6} b \ln \left (-c \,x^{3}+1\right ) c^{2}+2 b c \,x^{3}-b \ln \left (-c \,x^{3}+1\right )+2 a \right ) \ln \left (c \,x^{3}+1\right )}{12 x^{6}}+\frac {b^{2} c^{2} x^{6} \ln \left (-c \,x^{3}+1\right )^{2}+24 b^{2} c^{2} \ln \relax (x ) x^{6}+4 b \,c^{2} \ln \left (c \,x^{3}+1\right ) x^{6} a -4 b^{2} c^{2} \ln \left (c \,x^{3}+1\right ) x^{6}-4 b \,c^{2} \ln \left (c \,x^{3}-1\right ) x^{6} a -4 b^{2} c^{2} \ln \left (c \,x^{3}-1\right ) x^{6}+4 b^{2} c \,x^{3} \ln \left (-c \,x^{3}+1\right )-8 a b c \,x^{3}-b^{2} \ln \left (-c \,x^{3}+1\right )^{2}+4 b \ln \left (-c \,x^{3}+1\right ) a -4 a^{2}}{24 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.33, size = 175, normalized size = 1.99 \[ \frac {1}{6} \, {\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{6}}\right )} a b + \frac {1}{24} \, {\left ({\left (2 \, {\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right ) + 24 \, \log \relax (x)\right )} c^{2} + 4 \, {\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c \operatorname {artanh}\left (c x^{3}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac {a^{2}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.54, size = 278, normalized size = 3.16 \[ \frac {b^2\,c^2\,{\ln \left (c\,x^3+1\right )}^2}{24}-\frac {b^2\,c^2\,\ln \left (c\,x^3-1\right )}{6}-\frac {b^2\,c^2\,\ln \left (c\,x^3+1\right )}{6}-\frac {a^2}{6\,x^6}+\frac {b^2\,c^2\,{\ln \left (1-c\,x^3\right )}^2}{24}-\frac {b^2\,{\ln \left (c\,x^3+1\right )}^2}{24\,x^6}-\frac {b^2\,{\ln \left (1-c\,x^3\right )}^2}{24\,x^6}+b^2\,c^2\,\ln \relax (x)-\frac {a\,b\,c^2\,\ln \left (c\,x^3-1\right )}{6}+\frac {a\,b\,c^2\,\ln \left (c\,x^3+1\right )}{6}-\frac {a\,b\,c}{3\,x^3}-\frac {a\,b\,\ln \left (c\,x^3+1\right )}{6\,x^6}+\frac {a\,b\,\ln \left (1-c\,x^3\right )}{6\,x^6}-\frac {b^2\,c^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12}-\frac {b^2\,c\,\ln \left (c\,x^3+1\right )}{6\,x^3}+\frac {b^2\,c\,\ln \left (1-c\,x^3\right )}{6\,x^3}+\frac {b^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________