Optimal. Leaf size=139 \[ \frac {3}{4} b c^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2-\frac {3 b c \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 x^2}+\frac {1}{4} c^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3-\frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{4 x^4}+\frac {3}{2} b^2 c^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \log \left (2-\frac {2}{1+c x^2}\right )-\frac {3}{4} b^3 c^2 \text {PolyLog}\left (2,-1+\frac {2}{1+c x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6039, 6037,
6129, 6135, 6079, 2497, 6095} \begin {gather*} \frac {3}{2} b^2 c^2 \log \left (2-\frac {2}{c x^2+1}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac {3}{4} b c^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2+\frac {1}{4} c^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3-\frac {3 b c \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 x^2}-\frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{4 x^4}-\frac {3}{4} b^3 c^2 \text {Li}_2\left (\frac {2}{c x^2+1}-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2497
Rule 6037
Rule 6039
Rule 6079
Rule 6095
Rule 6129
Rule 6135
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{x^5} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{8 x^5}+\frac {3 b \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{8 x^5}-\frac {3 b^2 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{8 x^5}+\frac {b^3 \log ^3\left (1+c x^2\right )}{8 x^5}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{x^5} \, dx+\frac {1}{8} (3 b) \int \frac {\left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{x^5} \, dx-\frac {1}{8} \left (3 b^2\right ) \int \frac {\left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{x^5} \, dx+\frac {1}{8} b^3 \int \frac {\log ^3\left (1+c x^2\right )}{x^5} \, dx\\ &=\frac {1}{16} \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^3}{x^3} \, dx,x,x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{16} b^3 \text {Subst}\left (\int \frac {\log ^3(1+c x)}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^2 (1-c x)} \, dx,x,x^2\right )+\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^2 (1+c x)} \, dx,x,x^2\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}-\frac {1}{32} (3 b) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{32} \left (3 b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+c x^2\right )\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}-\frac {1}{32} (3 b) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{32} \left (3 b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+c x^2\right )-\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-c x^2\right )-\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+c x^2\right )\\ &=-\frac {3 b c \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 x^2}-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {3 b^3 c \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 x^2}-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^2\right )-\frac {1}{16} \left (3 b^2 c\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^2\right )-\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+c x^2\right )+\frac {1}{16} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+c x^2\right )-\frac {1}{32} \left (3 b c^2\right ) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x} \, dx,x,1-c x^2\right )+\frac {1}{32} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,1+c x^2\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {3 b c \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 x^2}+\frac {3}{32} b c^2 \log \left (c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {3 b^3 c \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 x^2}-\frac {3}{32} b^3 c^2 \log \left (-c x^2\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}-\frac {3}{16} b^3 c^2 \text {Li}_2\left (-c x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )+\frac {1}{16} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x^2\right )+\frac {1}{32} \left (3 c^2\right ) \text {Subst}\left (\int x^2 \, dx,x,2 a-b \log \left (1-c x^2\right )\right )+\frac {1}{16} \left (3 b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )+\frac {1}{32} \left (3 b^3 c^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log \left (1+c x^2\right )\right )+\frac {1}{16} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1+c x^2\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {3 b c \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 x^2}+\frac {3}{32} b c^2 \log \left (c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{32} c^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {3 b^3 c \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 x^2}-\frac {3}{32} b^3 c^2 \log \left (-c x^2\right ) \log ^2\left (1+c x^2\right )+\frac {1}{32} b^3 c^2 \log ^3\left (1+c x^2\right )-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}-\frac {3}{16} b^3 c^2 \text {Li}_2\left (-c x^2\right )+\frac {3}{16} b^3 c^2 \text {Li}_2\left (c x^2\right )-\frac {3}{16} b^2 c^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text {Li}_2\left (1-c x^2\right )-\frac {3}{16} b^3 c^2 \log \left (1+c x^2\right ) \text {Li}_2\left (1+c x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x^2\right )+\frac {1}{16} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+c x^2\right )\\ &=\frac {3}{4} a b^2 c^2 \log (x)-\frac {3 b c \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 x^2}+\frac {3}{32} b c^2 \log \left (c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{32} c^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 x^4}-\frac {3 b^3 c \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 x^2}-\frac {3}{32} b^3 c^2 \log \left (-c x^2\right ) \log ^2\left (1+c x^2\right )+\frac {1}{32} b^3 c^2 \log ^3\left (1+c x^2\right )-\frac {b^3 \log ^3\left (1+c x^2\right )}{32 x^4}-\frac {3}{16} b^3 c^2 \text {Li}_2\left (-c x^2\right )+\frac {3}{16} b^3 c^2 \text {Li}_2\left (c x^2\right )-\frac {3}{16} b^2 c^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text {Li}_2\left (1-c x^2\right )-\frac {3}{16} b^3 c^2 \log \left (1+c x^2\right ) \text {Li}_2\left (1+c x^2\right )-\frac {3}{16} b^3 c^2 \text {Li}_3\left (1-c x^2\right )+\frac {3}{16} b^3 c^2 \text {Li}_3\left (1+c x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2 \log (1+c x)}{x^3} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log ^2(1+c x)}{x^3} \, dx,x,x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 218, normalized size = 1.57 \begin {gather*} \frac {6 b^2 \left (-1+c x^2\right ) \left (a+a c x^2+b c x^2\right ) \tanh ^{-1}\left (c x^2\right )^2+2 b^3 \left (-1+c^2 x^4\right ) \tanh ^{-1}\left (c x^2\right )^3-6 b \tanh ^{-1}\left (c x^2\right ) \left (a^2+2 a b c x^2-2 b^2 c^2 x^4 \log \left (1-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )\right )+a \left (-2 a^2-6 a b c x^2-3 a b c^2 x^4 \log \left (1-c x^2\right )+3 a b c^2 x^4 \log \left (1+c x^2\right )+12 b^2 c^2 x^4 \log \left (\frac {c x^2}{\sqrt {1-c^2 x^4}}\right )\right )-6 b^3 c^2 x^4 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{3}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{3}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^3}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________