Optimal. Leaf size=104 \[ -\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2}+\frac {2 b \log \left (2-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^2}-\frac {b^2 \text {Li}_2\left (\frac {2}{c+d x+1}-1\right )}{d e^2} \]
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Rubi [A] time = 0.18, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6107, 12, 5916, 5988, 5932, 2447} \[ -\frac {b^2 \text {PolyLog}\left (2,\frac {2}{c+d x+1}-1\right )}{d e^2}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2}+\frac {2 b \log \left (2-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2447
Rule 5916
Rule 5932
Rule 5988
Rule 6107
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{(c e+d e x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x (1+x)} \, dx,x,c+d x\right )}{d e^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {2 b \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^2}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (2-\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {2 b \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^2}-\frac {b^2 \text {Li}_2\left (-1+\frac {2}{1+c+d x}\right )}{d e^2}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 126, normalized size = 1.21 \[ \frac {a \left (2 b (c+d x) \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )-a\right )+2 b \tanh ^{-1}(c+d x) \left (b (c+d x) \log \left (1-e^{-2 \tanh ^{-1}(c+d x)}\right )-a\right )-b^2 (c+d x) \text {Li}_2\left (e^{-2 \tanh ^{-1}(c+d x)}\right )+b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2}{d e^2 (c+d x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {artanh}\left (d x + c\right ) + a^{2}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 396, normalized size = 3.81 \[ -\frac {a^{2}}{d \,e^{2} \left (d x +c \right )}-\frac {b^{2} \arctanh \left (d x +c \right )^{2}}{d \,e^{2} \left (d x +c \right )}+\frac {2 b^{2} \ln \left (d x +c \right ) \arctanh \left (d x +c \right )}{d \,e^{2}}-\frac {b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c -1\right )}{d \,e^{2}}-\frac {b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c +1\right )}{d \,e^{2}}-\frac {b^{2} \ln \left (d x +c -1\right )^{2}}{4 d \,e^{2}}+\frac {b^{2} \dilog \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d \,e^{2}}+\frac {b^{2} \ln \left (d x +c -1\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,e^{2}}+\frac {b^{2} \ln \left (d x +c +1\right )^{2}}{4 d \,e^{2}}-\frac {b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{2 d \,e^{2}}+\frac {b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,e^{2}}-\frac {b^{2} \dilog \left (d x +c \right )}{d \,e^{2}}-\frac {b^{2} \dilog \left (d x +c +1\right )}{d \,e^{2}}-\frac {b^{2} \ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{d \,e^{2}}-\frac {2 a b \arctanh \left (d x +c \right )}{d \,e^{2} \left (d x +c \right )}+\frac {2 a b \ln \left (d x +c \right )}{d \,e^{2}}-\frac {a b \ln \left (d x +c -1\right )}{d \,e^{2}}-\frac {a b \ln \left (d x +c +1\right )}{d \,e^{2}} \]
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 \[ -{\left (d {\left (\frac {\log \left (d x + c + 1\right )}{d^{2} e^{2}} - \frac {2 \, \log \left (d x + c\right )}{d^{2} e^{2}} + \frac {\log \left (d x + c - 1\right )}{d^{2} e^{2}}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{d^{2} e^{2} x + c d e^{2}}\right )} a b - \frac {1}{4} \, b^{2} {\left (\frac {\log \left (-d x - c + 1\right )^{2}}{d^{2} e^{2} x + c d e^{2}} + \int -\frac {{\left (d x + c - 1\right )} \log \left (d x + c + 1\right )^{2} + 2 \, {\left (d x - {\left (d x + c - 1\right )} \log \left (d x + c + 1\right ) + c\right )} \log \left (-d x - c + 1\right )}{d^{3} e^{2} x^{3} + c^{3} e^{2} - c^{2} e^{2} + {\left (3 \, c d^{2} e^{2} - d^{2} e^{2}\right )} x^{2} + {\left (3 \, c^{2} d e^{2} - 2 \, c d e^{2}\right )} x}\,{d x}\right )} - \frac {a^{2}}{d^{2} e^{2} x + c d e^{2}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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