Optimal. Leaf size=178 \[ \frac {1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+6 a c^2 d^4 x+4 a c d^4 \log (x)+\frac {1}{6} b c^3 d^4 x^2+\frac {8}{3} b c d^4 \log \left (1-c^2 x^2\right )+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)-2 b c d^4 \text {Li}_2(-c x)+2 b c d^4 \text {Li}_2(c x)+b c d^4 \log (x)-2 b c d^4 \tanh ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5940, 5910, 260, 5916, 266, 36, 29, 31, 5912, 321, 206, 43} \[ -2 b c d^4 \text {PolyLog}(2,-c x)+2 b c d^4 \text {PolyLog}(2,c x)+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+6 a c^2 d^4 x+4 a c d^4 \log (x)+\frac {1}{6} b c^3 d^4 x^2+\frac {8}{3} b c d^4 \log \left (1-c^2 x^2\right )+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)+b c d^4 \log (x)-2 b c d^4 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 43
Rule 206
Rule 260
Rule 266
Rule 321
Rule 5910
Rule 5912
Rule 5916
Rule 5940
Rubi steps
\begin {align*} \int \frac {(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 c^3 d^4 x \left (a+b \tanh ^{-1}(c x)\right )+c^4 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (4 c d^4\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+\left (6 c^2 d^4\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (4 c^3 d^4\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=6 a c^2 d^4 x-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)-2 b c d^4 \text {Li}_2(-c x)+2 b c d^4 \text {Li}_2(c x)+\left (b c d^4\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (6 b c^2 d^4\right ) \int \tanh ^{-1}(c x) \, dx-\left (2 b c^4 d^4\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{3} \left (b c^5 d^4\right ) \int \frac {x^3}{1-c^2 x^2} \, dx\\ &=6 a c^2 d^4 x+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)-2 b c d^4 \text {Li}_2(-c x)+2 b c d^4 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )-\left (2 b c^2 d^4\right ) \int \frac {1}{1-c^2 x^2} \, dx-\left (6 b c^3 d^4\right ) \int \frac {x}{1-c^2 x^2} \, dx-\frac {1}{6} \left (b c^5 d^4\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )\\ &=6 a c^2 d^4 x+2 b c^2 d^4 x-2 b c d^4 \tanh ^{-1}(c x)+6 b c^2 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)+3 b c d^4 \log \left (1-c^2 x^2\right )-2 b c d^4 \text {Li}_2(-c x)+2 b c d^4 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b c^3 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{6} \left (b c^5 d^4\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=6 a c^2 d^4 x+2 b c^2 d^4 x+\frac {1}{6} b c^3 d^4 x^2-2 b c d^4 \tanh ^{-1}(c x)+6 b c^2 d^4 x \tanh ^{-1}(c x)-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)+b c d^4 \log (x)+\frac {8}{3} b c d^4 \log \left (1-c^2 x^2\right )-2 b c d^4 \text {Li}_2(-c x)+2 b c d^4 \text {Li}_2(c x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 194, normalized size = 1.09 \[ \frac {d^4 \left (2 a c^4 x^4+12 a c^3 x^3+36 a c^2 x^2+24 a c x \log (x)-6 a+2 b c^4 x^4 \tanh ^{-1}(c x)+b c^3 x^3+12 b c^3 x^3 \tanh ^{-1}(c x)+12 b c^2 x^2+15 b c x \log \left (1-c^2 x^2\right )+b c x \log \left (c^2 x^2-1\right )+36 b c^2 x^2 \tanh ^{-1}(c x)-12 b c x \text {Li}_2(-c x)+12 b c x \text {Li}_2(c x)+6 b c x \log (c x)+6 b c x \log (1-c x)-6 b c x \log (c x+1)-6 b \tanh ^{-1}(c x)\right )}{6 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{4} d^{4} x^{4} + 4 \, a c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{4} x^{2} + 4 \, a c d^{4} x + a d^{4} + {\left (b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 4 \, b c d^{4} x + b d^{4}\right )} \operatorname {artanh}\left (c x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d x + d\right )}^{4} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 229, normalized size = 1.29 \[ \frac {d^{4} a \,c^{4} x^{3}}{3}+2 d^{4} a \,c^{3} x^{2}+6 a \,c^{2} d^{4} x +4 c \,d^{4} a \ln \left (c x \right )-\frac {d^{4} a}{x}+\frac {d^{4} b \arctanh \left (c x \right ) c^{4} x^{3}}{3}+2 d^{4} b \arctanh \left (c x \right ) c^{3} x^{2}+6 b \,c^{2} d^{4} x \arctanh \left (c x \right )+4 c \,d^{4} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{4} b \arctanh \left (c x \right )}{x}-2 c \,d^{4} b \dilog \left (c x \right )-2 c \,d^{4} b \dilog \left (c x +1\right )-2 c \,d^{4} b \ln \left (c x \right ) \ln \left (c x +1\right )+\frac {b \,c^{3} d^{4} x^{2}}{6}+2 b \,c^{2} d^{4} x +c \,d^{4} b \ln \left (c x \right )+\frac {11 c \,d^{4} b \ln \left (c x -1\right )}{3}+\frac {5 c \,d^{4} b \ln \left (c x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 281, normalized size = 1.58 \[ \frac {1}{3} \, a c^{4} d^{4} x^{3} + 2 \, a c^{3} d^{4} x^{2} + \frac {1}{6} \, b c^{3} d^{4} x^{2} + 6 \, a c^{2} d^{4} x + 2 \, b c^{2} d^{4} x + 3 \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c d^{4} - 2 \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b c d^{4} + 2 \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b c d^{4} - \frac {5}{6} \, b c d^{4} \log \left (c x + 1\right ) + \frac {7}{6} \, b c d^{4} \log \left (c x - 1\right ) + 4 \, a c d^{4} \log \relax (x) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b d^{4} - \frac {a d^{4}}{x} + \frac {1}{6} \, {\left (b c^{4} d^{4} x^{3} + 6 \, b c^{3} d^{4} x^{2}\right )} \log \left (c x + 1\right ) - \frac {1}{6} \, {\left (b c^{4} d^{4} x^{3} + 6 \, b c^{3} d^{4} x^{2}\right )} \log \left (-c x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^4}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{4} \left (\int 6 a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int \frac {4 a c}{x}\, dx + \int 4 a c^{3} x\, dx + \int a c^{4} x^{2}\, dx + \int 6 b c^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {4 b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 4 b c^{3} x \operatorname {atanh}{\left (c x \right )}\, dx + \int b c^{4} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________