Optimal. Leaf size=124 \[ \frac {b}{2 d^2 (1+c x)}-\frac {b \tanh ^{-1}(c x)}{2 d^2}+\frac {a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \text {PolyLog}(2,-c x)}{2 d^2}+\frac {b \text {PolyLog}(2,c x)}{2 d^2}-\frac {b \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 124, 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 = {6087, 6031,
6063, 641, 46, 213, 6055, 2449, 2352} \begin {gather*} \frac {a+b \tanh ^{-1}(c x)}{d^2 (c x+1)}+\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {a \log (x)}{d^2}-\frac {b \text {Li}_2(-c x)}{2 d^2}+\frac {b \text {Li}_2(c x)}{2 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 d^2}+\frac {b}{2 d^2 (c x+1)}-\frac {b \tanh ^{-1}(c x)}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 213
Rule 641
Rule 2352
Rule 2449
Rule 6031
Rule 6055
Rule 6063
Rule 6087
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x (d+c d x)^2} \, dx &=\int \left (\frac {a+b \tanh ^{-1}(c x)}{d^2 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)^2}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx}{d^2}-\frac {c \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{d^2}-\frac {c \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{d^2}\\ &=\frac {a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \text {Li}_2(-c x)}{2 d^2}+\frac {b \text {Li}_2(c x)}{2 d^2}-\frac {(b c) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{d^2}-\frac {(b c) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=\frac {a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \text {Li}_2(-c x)}{2 d^2}+\frac {b \text {Li}_2(c x)}{2 d^2}-\frac {b \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{d^2}-\frac {(b c) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{d^2}\\ &=\frac {a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \text {Li}_2(-c x)}{2 d^2}+\frac {b \text {Li}_2(c x)}{2 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {(b c) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=\frac {b}{2 d^2 (1+c x)}+\frac {a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \text {Li}_2(-c x)}{2 d^2}+\frac {b \text {Li}_2(c x)}{2 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 d^2}+\frac {(b c) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {b}{2 d^2 (1+c x)}-\frac {b \tanh ^{-1}(c x)}{2 d^2}+\frac {a+b \tanh ^{-1}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \text {Li}_2(-c x)}{2 d^2}+\frac {b \text {Li}_2(c x)}{2 d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 101, normalized size = 0.81 \begin {gather*} \frac {\frac {4 a}{1+c x}+4 a \log (x)-4 a \log (1+c x)+b \left (\cosh \left (2 \tanh ^{-1}(c x)\right )-2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+2 \tanh ^{-1}(c x) \left (\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )\right )}{4 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 221, normalized size = 1.78
method | result | size |
risch | \(\frac {a \ln \left (-c x \right )}{d^{2}}-\frac {a}{d^{2} \left (-c x -1\right )}-\frac {a \ln \left (-c x -1\right )}{d^{2}}-\frac {b \ln \left (-c x -1\right )}{4 d^{2}}-\frac {b \ln \left (-c x +1\right ) c x}{4 d^{2} \left (-c x -1\right )}+\frac {b \ln \left (-c x +1\right )}{4 d^{2} \left (-c x -1\right )}+\frac {\dilog \left (-c x +1\right ) b}{2 d^{2}}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{2}}-\frac {b \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}-\frac {b \ln \left (c x +1\right )^{2}}{4 d^{2}}-\frac {b \dilog \left (c x +1\right )}{2 d^{2}}+\frac {b \ln \left (c x +1\right )}{2 d^{2} \left (c x +1\right )}+\frac {b}{2 d^{2} \left (c x +1\right )}\) | \(220\) |
derivativedivides | \(\frac {a \ln \left (c x \right )}{d^{2}}+\frac {a}{d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{d^{2}}+\frac {b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{2}}+\frac {b \arctanh \left (c x \right )}{d^{2} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{2}}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{2}}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b \ln \left (c x +1\right )^{2}}{4 d^{2}}+\frac {b \ln \left (c x -1\right )}{4 d^{2}}+\frac {b}{2 d^{2} \left (c x +1\right )}-\frac {b \ln \left (c x +1\right )}{4 d^{2}}-\frac {b \dilog \left (c x \right )}{2 d^{2}}-\frac {b \dilog \left (c x +1\right )}{2 d^{2}}-\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{2}}\) | \(221\) |
default | \(\frac {a \ln \left (c x \right )}{d^{2}}+\frac {a}{d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{d^{2}}+\frac {b \arctanh \left (c x \right ) \ln \left (c x \right )}{d^{2}}+\frac {b \arctanh \left (c x \right )}{d^{2} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d^{2}}-\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d^{2}}+\frac {b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b \ln \left (c x +1\right )^{2}}{4 d^{2}}+\frac {b \ln \left (c x -1\right )}{4 d^{2}}+\frac {b}{2 d^{2} \left (c x +1\right )}-\frac {b \ln \left (c x +1\right )}{4 d^{2}}-\frac {b \dilog \left (c x \right )}{2 d^{2}}-\frac {b \dilog \left (c x +1\right )}{2 d^{2}}-\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d^{2}}\) | \(221\) |
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*} \frac {\int \frac {a}{c^{2} x^{3} + 2 c x^{2} + x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{3} + 2 c x^{2} + x}\, dx}{d^{2}} \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 {a+b\,\mathrm {atanh}\left (c\,x\right )}{x\,{\left (d+c\,d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________