Optimal. Leaf size=168 \[ -\frac {b \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e}+\frac {b \text {Li}_2\left (\frac {2}{-c-d x+1}-1\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{-c-d x+1}\right )}{2 d e}-\frac {b^2 \text {Li}_3\left (\frac {2}{-c-d x+1}-1\right )}{2 d e} \]
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Rubi [A] time = 0.30, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6107, 12, 5914, 6052, 5948, 6058, 6610} \[ -\frac {b \text {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e}+\frac {b \text {PolyLog}\left (2,\frac {2}{-c-d x+1}-1\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{2 d e}-\frac {b^2 \text {PolyLog}\left (3,\frac {2}{-c-d x+1}-1\right )}{2 d e}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 5914
Rule 5948
Rule 6052
Rule 6058
Rule 6107
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{c e+d e x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c-d x}\right )}{d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (1-\frac {2}{1-x}\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c-d x}\right )}{d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right ) \log \left (2-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right ) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c-d x}\right )}{d e}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d e}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c-d x}\right )}{d e}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-1+\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c-d x}\right )}{d e}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d e}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c-d x}\right )}{d e}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-c-d x}\right )}{2 d e}-\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1-c-d x}\right )}{2 d e}\\ \end {align*}
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Mathematica [C] time = 0.39, size = 424, normalized size = 2.52 \[ \frac {a^2 \log (c+d x)-\frac {1}{4} a b \left (4 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c+d x)}\right )+4 \text {Li}_2\left (-e^{2 \tanh ^{-1}(c+d x)}\right )-4 i \pi \log \left (\frac {2}{\sqrt {1-(c+d x)^2}}\right )-8 \tanh ^{-1}(c+d x)^2-4 i \pi \tanh ^{-1}(c+d x)-8 \tanh ^{-1}(c+d x) \log \left (1-e^{-2 \tanh ^{-1}(c+d x)}\right )+8 \tanh ^{-1}(c+d x) \log \left (e^{2 \tanh ^{-1}(c+d x)}+1\right )-8 \log \left (\frac {2}{\sqrt {1-(c+d x)^2}}\right ) \tanh ^{-1}(c+d x)+8 \log \left (\frac {2 i (c+d x)}{\sqrt {1-(c+d x)^2}}\right ) \tanh ^{-1}(c+d x)+4 i \pi \log \left (e^{2 \tanh ^{-1}(c+d x)}+1\right )+\pi ^2\right )+2 a b \left (-\log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\log \left (\frac {i (c+d x)}{\sqrt {1-(c+d x)^2}}\right )\right ) \tanh ^{-1}(c+d x)+b^2 \left (\tanh ^{-1}(c+d x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )+\tanh ^{-1}(c+d x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c+d x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \tanh ^{-1}(c+d x)}\right )-\frac {2}{3} \tanh ^{-1}(c+d x)^3-\tanh ^{-1}(c+d x)^2 \log \left (e^{-2 \tanh ^{-1}(c+d x)}+1\right )+\tanh ^{-1}(c+d x)^2 \log \left (1-e^{2 \tanh ^{-1}(c+d x)}\right )+\frac {i \pi ^3}{24}\right )}{d e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {artanh}\left (d x + c\right ) + a^{2}}{d e x + c e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 893, normalized size = 5.32 \[ \frac {a^{2} \ln \left (d x +c \right )}{d e}+\frac {b^{2} \ln \left (d x +c \right ) \arctanh \left (d x +c \right )^{2}}{d e}-\frac {b^{2} \arctanh \left (d x +c \right ) \polylog \left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {b^{2} \polylog \left (3, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{2 d e}-\frac {b^{2} \arctanh \left (d x +c \right )^{2} \ln \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )}{d e}+\frac {b^{2} \arctanh \left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{d e}+\frac {2 b^{2} \arctanh \left (d x +c \right ) \polylog \left (2, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{d e}-\frac {2 b^{2} \polylog \left (3, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{d e}+\frac {b^{2} \arctanh \left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{d e}+\frac {2 b^{2} \arctanh \left (d x +c \right ) \polylog \left (2, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{d e}-\frac {2 b^{2} \polylog \left (3, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{d e}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )}{1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}}\right )^{2} \arctanh \left (d x +c \right )^{2}}{2 d e}+\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )}{1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}}\right )^{3} \arctanh \left (d x +c \right )^{2}}{2 d e}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )}{1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}}\right )^{2} \arctanh \left (d x +c \right )^{2}}{2 d e}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )}{1+\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}}\right ) \arctanh \left (d x +c \right )^{2}}{2 d e}+\frac {2 a b \ln \left (d x +c \right ) \arctanh \left (d x +c \right )}{d e}-\frac {a b \dilog \left (d x +c \right )}{d e}-\frac {a b \dilog \left (d x +c +1\right )}{d e}-\frac {a b \ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \log \left (d e x + c e\right )}{d e} + \int \frac {b^{2} {\left (\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )\right )}^{2}}{4 \, {\left (d e x + c e\right )}} + \frac {a b {\left (\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )\right )}}{d e x + c e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{c\,e+d\,e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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