Optimal. Leaf size=145 \[ -\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c d}-\frac {a x}{c^2 d}+\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 c^3 d}-\frac {b \tanh ^{-1}(c x)}{2 c^3 d}+\frac {b x}{2 c^2 d}-\frac {b x \tanh ^{-1}(c x)}{c^2 d}-\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5930, 5916, 321, 206, 5910, 260, 5918, 2402, 2315} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c^3 d}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c d}-\frac {a x}{c^2 d}-\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b x}{2 c^2 d}-\frac {b x \tanh ^{-1}(c x)}{c^2 d}-\frac {b \tanh ^{-1}(c x)}{2 c^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5930
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{d+c d x} \, dx &=-\frac {\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+c d x} \, dx}{c}+\frac {\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c d}\\ &=\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c d}+\frac {\int \frac {a+b \tanh ^{-1}(c x)}{d+c d x} \, dx}{c^2}-\frac {b \int \frac {x^2}{1-c^2 x^2} \, dx}{2 d}-\frac {\int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2 d}\\ &=-\frac {a x}{c^2 d}+\frac {b x}{2 c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c d}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d}-\frac {b \int \frac {1}{1-c^2 x^2} \, dx}{2 c^2 d}-\frac {b \int \tanh ^{-1}(c x) \, dx}{c^2 d}+\frac {b \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^2 d}\\ &=-\frac {a x}{c^2 d}+\frac {b x}{2 c^2 d}-\frac {b \tanh ^{-1}(c x)}{2 c^3 d}-\frac {b x \tanh ^{-1}(c x)}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c d}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{c^3 d}+\frac {b \int \frac {x}{1-c^2 x^2} \, dx}{c d}\\ &=-\frac {a x}{c^2 d}+\frac {b x}{2 c^2 d}-\frac {b \tanh ^{-1}(c x)}{2 c^3 d}-\frac {b x \tanh ^{-1}(c x)}{c^2 d}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c d}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{c^3 d}-\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 97, normalized size = 0.67 \[ \frac {a c^2 x^2-2 a c x+2 a \log (c x+1)-b \log \left (1-c^2 x^2\right )+b \tanh ^{-1}(c x) \left (c^2 x^2-2 c x-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-1\right )+b \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+b c x}{2 c^3 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {artanh}\left (c x\right ) + a x^{2}}{c d x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{2}}{c d x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 213, normalized size = 1.47 \[ \frac {a \,x^{2}}{2 c d}-\frac {a x}{c^{2} d}+\frac {a \ln \left (c x +1\right )}{c^{3} d}+\frac {b \arctanh \left (c x \right ) x^{2}}{2 c d}-\frac {b x \arctanh \left (c x \right )}{c^{2} d}+\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c^{3} d}-\frac {b \ln \left (c x +1\right )^{2}}{4 c^{3} d}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 c^{3} d}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c^{3} d}-\frac {b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 c^{3} d}+\frac {b x}{2 c^{2} d}+\frac {b}{2 c^{3} d}-\frac {3 b \ln \left (c x +1\right )}{4 c^{3} d}-\frac {b \ln \left (c x -1\right )}{4 c^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (c^{3} {\left (\frac {x^{2}}{c^{4} d} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{6} d}\right )} + 8 \, c^{3} \int \frac {x^{3} \log \left (c x + 1\right )}{2 \, {\left (c^{4} d x^{2} - c^{2} d\right )}}\,{d x} - c^{2} {\left (\frac {2 \, x}{c^{4} d} - \frac {\log \left (c x + 1\right )}{c^{5} d} + \frac {\log \left (c x - 1\right )}{c^{5} d}\right )} - 8 \, c^{2} \int \frac {x^{2} \log \left (c x + 1\right )}{2 \, {\left (c^{4} d x^{2} - c^{2} d\right )}}\,{d x} + 8 \, c \int \frac {x \log \left (c x + 1\right )}{2 \, {\left (c^{4} d x^{2} - c^{2} d\right )}}\,{d x} - \frac {2 \, {\left (c^{2} x^{2} - 2 \, c x + 2 \, \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c^{3} d} - \frac {2 \, \log \left (2 \, c^{4} d x^{2} - 2 \, c^{2} d\right )}{c^{3} d} + 8 \, \int \frac {\log \left (c x + 1\right )}{2 \, {\left (c^{4} d x^{2} - c^{2} d\right )}}\,{d x}\right )} b + \frac {1}{2} \, a {\left (\frac {c x^{2} - 2 \, x}{c^{2} d} + \frac {2 \, \log \left (c x + 1\right )}{c^{3} d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+c\,d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{2}}{c x + 1}\, dx + \int \frac {b x^{2} \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________