Optimal. Leaf size=149 \[ -\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (c x+1)}+\frac {2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}+\frac {a x}{c^2 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{c^3 d^2}-\frac {b}{2 c^3 d^2 (c x+1)}+\frac {b \tanh ^{-1}(c x)}{2 c^3 d^2}+\frac {b x \tanh ^{-1}(c x)}{c^2 d^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5940, 5910, 260, 5926, 627, 44, 207, 5918, 2402, 2315} \[ -\frac {b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{c^3 d^2}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (c x+1)}+\frac {2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}+\frac {a x}{c^2 d^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2}-\frac {b}{2 c^3 d^2 (c x+1)}+\frac {b x \tanh ^{-1}(c x)}{c^2 d^2}+\frac {b \tanh ^{-1}(c x)}{2 c^3 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 207
Rule 260
Rule 627
Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5926
Rule 5940
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{(d+c d x)^2} \, dx &=\int \left (\frac {a+b \tanh ^{-1}(c x)}{c^2 d^2}+\frac {a+b \tanh ^{-1}(c x)}{c^2 d^2 (1+c x)^2}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2 (1+c x)}\right ) \, dx\\ &=\frac {\int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2 d^2}+\frac {\int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^2 d^2}-\frac {2 \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{c^2 d^2}\\ &=\frac {a x}{c^2 d^2}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}+\frac {b \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^2 d^2}+\frac {b \int \tanh ^{-1}(c x) \, dx}{c^2 d^2}-\frac {(2 b) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d^2}\\ &=\frac {a x}{c^2 d^2}+\frac {b x \tanh ^{-1}(c x)}{c^2 d^2}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c^3 d^2}+\frac {b \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c^2 d^2}-\frac {b \int \frac {x}{1-c^2 x^2} \, dx}{c d^2}\\ &=\frac {a x}{c^2 d^2}+\frac {b x \tanh ^{-1}(c x)}{c^2 d^2}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d^2}+\frac {b \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^2 d^2}\\ &=\frac {a x}{c^2 d^2}-\frac {b}{2 c^3 d^2 (1+c x)}+\frac {b x \tanh ^{-1}(c x)}{c^2 d^2}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d^2}-\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{2 c^2 d^2}\\ &=\frac {a x}{c^2 d^2}-\frac {b}{2 c^3 d^2 (1+c x)}+\frac {b \tanh ^{-1}(c x)}{2 c^3 d^2}+\frac {b x \tanh ^{-1}(c x)}{c^2 d^2}-\frac {a+b \tanh ^{-1}(c x)}{c^3 d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^3 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.58, size = 121, normalized size = 0.81 \[ \frac {4 a c x-\frac {4 a}{c x+1}-8 a \log (c x+1)+b \left (2 \log \left (1-c^2 x^2\right )-4 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (2 c x+4 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )}{4 c^3 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {artanh}\left (c x\right ) + a x^{2}}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{2}}{{\left (c d x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 216, normalized size = 1.45 \[ \frac {a x}{c^{2} d^{2}}-\frac {a}{c^{3} d^{2} \left (c x +1\right )}-\frac {2 a \ln \left (c x +1\right )}{c^{3} d^{2}}+\frac {b x \arctanh \left (c x \right )}{c^{2} d^{2}}-\frac {b \arctanh \left (c x \right )}{c^{3} d^{2} \left (c x +1\right )}-\frac {2 b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c^{3} d^{2}}+\frac {b \ln \left (c x +1\right )^{2}}{2 c^{3} d^{2}}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{c^{3} d^{2}}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{c^{3} d^{2}}+\frac {b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{c^{3} d^{2}}-\frac {b}{2 c^{3} d^{2} \left (c x +1\right )}+\frac {3 b \ln \left (c x +1\right )}{4 c^{3} d^{2}}+\frac {b \ln \left (c x -1\right )}{4 c^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, {\left (c^{3} {\left (\frac {2}{c^{7} d^{2} x + c^{6} d^{2}} - \frac {4 \, x}{c^{5} d^{2}} + \frac {5 \, \log \left (c x + 1\right )}{c^{6} d^{2}} - \frac {\log \left (c x - 1\right )}{c^{6} d^{2}}\right )} - 4 \, c^{3} \int \frac {x^{3} \log \left (c x + 1\right )}{c^{5} d^{2} x^{3} + c^{4} d^{2} x^{2} - c^{3} d^{2} x - c^{2} d^{2}}\,{d x} - 2 \, c^{2} {\left (\frac {2}{c^{6} d^{2} x + c^{5} d^{2}} + \frac {3 \, \log \left (c x + 1\right )}{c^{5} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{5} d^{2}}\right )} + 12 \, c^{2} \int \frac {x^{2} \log \left (c x + 1\right )}{c^{5} d^{2} x^{3} + c^{4} d^{2} x^{2} - c^{3} d^{2} x - c^{2} d^{2}}\,{d x} + 16 \, c \int \frac {x \log \left (c x + 1\right )}{c^{5} d^{2} x^{3} + c^{4} d^{2} x^{2} - c^{3} d^{2} x - c^{2} d^{2}}\,{d x} + \frac {4 \, {\left (c^{2} x^{2} + c x - 2 \, {\left (c x + 1\right )} \log \left (c x + 1\right ) - 1\right )} \log \left (-c x + 1\right )}{c^{4} d^{2} x + c^{3} d^{2}} + \frac {2}{c^{4} d^{2} x + c^{3} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{3} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{3} d^{2}} + 8 \, \int \frac {\log \left (c x + 1\right )}{c^{5} d^{2} x^{3} + c^{4} d^{2} x^{2} - c^{3} d^{2} x - c^{2} d^{2}}\,{d x}\right )} b - a {\left (\frac {1}{c^{4} d^{2} x + c^{3} d^{2}} - \frac {x}{c^{2} d^{2}} + \frac {2 \, \log \left (c x + 1\right )}{c^{3} d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{2}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b x^{2} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________