Optimal. Leaf size=480 \[ \frac {2 b c^3 d \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d+e)^2}-\frac {b c^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d-e)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {b^2 c^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{4 e (c d-e)^2}+\frac {b^2 c^2 \log (1-c x)}{2 (c d-e) (c d+e)^2}-\frac {b^2 c^2 \log (c x+1)}{2 (c d-e)^2 (c d+e)}+\frac {b^2 c^2 e \log (d+e x)}{(c d-e)^2 (c d+e)^2} \]
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Rubi [A] time = 0.50, antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5928, 5918, 2402, 2315, 5926, 706, 31, 633, 5920, 2447} \[ -\frac {b^2 c^3 d \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^3 d \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {b^2 c^2 \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{4 e (c d-e)^2}+\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {2 b c^3 d \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b c^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d+e)^2}-\frac {b c^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 e (c d-e)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (1-c x)}{2 (c d-e) (c d+e)^2}-\frac {b^2 c^2 \log (c x+1)}{2 (c d-e)^2 (c d+e)}+\frac {b^2 c^2 e \log (d+e x)}{(c d-e)^2 (c d+e)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 706
Rule 2315
Rule 2402
Rule 2447
Rule 5918
Rule 5920
Rule 5926
Rule 5928
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {(b c) \int \left (-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 (c d+e)^2 (-1+c x)}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 (c d-e)^2 (1+c x)}+\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )}{(-c d+e) (c d+e) (d+e x)^2}-\frac {2 c^2 d e^2 \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e)^2 (c d+e)^2 (d+e x)}\right ) \, dx}{e}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{2 (c d-e)^2 e}-\frac {\left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c x} \, dx}{2 e (c d+e)^2}-\frac {\left (2 b c^3 d e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{(c d-e)^2 (c d+e)^2}+\frac {(b c e) \int \frac {a+b \tanh ^{-1}(c x)}{(d+e x)^2} \, dx}{(-c d+e) (c d+e)}\\ &=\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)^2}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 e}+\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {\left (b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e)^2 e}-\frac {\left (2 b^2 c^4 d\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}+\frac {\left (2 b^2 c^4 d\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{2 e (c d+e)^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(d+e x) \left (1-c^2 x^2\right )} \, dx}{(-c d+e) (c d+e)}\\ &=\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)^2}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 e}+\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 (c d-e)^2 e}+\frac {\left (b^2 c^2\right ) \int \frac {-c^2 d+c^2 e x}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (2 b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 e (c d+e)^2}+\frac {\left (b^2 c^2 e^2\right ) \int \frac {1}{d+e x} \, dx}{(c d-e)^2 (c d+e)^2}\\ &=\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)^2}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 e}+\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^2 e \log (d+e x)}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {b^2 c^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {\left (b^2 c^4\right ) \int \frac {1}{c-c^2 x} \, dx}{2 (c d-e) (c d+e)^2}+\frac {\left (b^2 c^4\right ) \int \frac {1}{-c-c^2 x} \, dx}{2 (c d-e)^2 (c d+e)}\\ &=\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{(c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)^2}+\frac {b^2 c^2 \log (1-c x)}{2 (c d-e) (c d+e)^2}-\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 e}+\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {b^2 c^2 \log (1+c x)}{2 (c d-e)^2 (c d+e)}+\frac {b^2 c^2 e \log (d+e x)}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {b^2 c^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}\\ \end {align*}
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Mathematica [C] time = 7.06, size = 470, normalized size = 0.98 \[ -\frac {a^2}{2 e (d+e x)^2}-\frac {a b c^2 \left (\frac {\frac {2 e \left (c^2 \left (-d^2\right )+2 c^2 d (d+e x) \log (c (d+e x))+e^2\right )}{c (c d+e)^2 (d+e x)}-\log (c x+1)}{(e-c d)^2}+\frac {\log (1-c x)}{(c d+e)^2}+\frac {2 \tanh ^{-1}(c x)}{(c d+c e x)^2}\right )}{2 e}+\frac {b^2 c^2 \left (\frac {2 c d \left (-i \pi \left (\tanh ^{-1}(c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )\right )+\text {Li}_2\left (e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+\tanh ^{-1}(c x)\right )+i \pi \log \left (e^{2 \tanh ^{-1}(c x)}+1\right )\right )}{c^2 d^2-e^2}-\frac {2 \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )}}{e \sqrt {1-\frac {c^2 d^2}{e^2}}}-\frac {e \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^2}{c^2 (d+e x)^2}+\frac {2 e \left (c d \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right )-e \tanh ^{-1}(c x)\right )}{c^3 d^3-c d e^2}+\frac {2 x \tanh ^{-1}(c x) \left (c d \tanh ^{-1}(c x)-e\right )}{c d (d+e x)}\right )}{2 (c d-e) (c d+e)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 824, normalized size = 1.72 \[ \frac {c^{2} a b}{\left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}+\frac {c^{2} a b \ln \left (c x +1\right )}{2 e \left (c d -e \right )^{2}}+\frac {c^{2} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{4 e \left (c d +e \right )^{2}}+\frac {c^{2} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{4 e \left (c d -e \right )^{2}}-\frac {c^{2} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{4 e \left (c d -e \right )^{2}}-\frac {c^{2} a b \ln \left (c x -1\right )}{2 e \left (c d +e \right )^{2}}+\frac {c^{2} b^{2} e \ln \left (c x e +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}+\frac {c^{2} b^{2} \ln \left (c x -1\right )}{\left (c d +e \right ) \left (c d -e \right ) \left (2 c d +2 e \right )}-\frac {c^{2} b^{2} \ln \left (c x +1\right )}{\left (c d +e \right ) \left (c d -e \right ) \left (2 c d -2 e \right )}+\frac {c^{2} b^{2} \arctanh \left (c x \right )}{\left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}-\frac {c^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{2 e \left (c d +e \right )^{2}}+\frac {c^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{2 e \left (c d -e \right )^{2}}-\frac {c^{2} a b \arctanh \left (c x \right )}{\left (c x e +c d \right )^{2} e}-\frac {c^{3} b^{2} d \dilog \left (\frac {c x e -e}{-c d -e}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}+\frac {c^{3} b^{2} d \dilog \left (\frac {c x e +e}{-c d +e}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {c^{2} a^{2}}{2 \left (c x e +c d \right )^{2} e}-\frac {2 c^{3} a b d \ln \left (c x e +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {c^{3} b^{2} d \ln \left (c x e +c d \right ) \ln \left (\frac {c x e -e}{-c d -e}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {2 c^{3} b^{2} \arctanh \left (c x \right ) d \ln \left (c x e +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}+\frac {c^{3} b^{2} d \ln \left (c x e +c d \right ) \ln \left (\frac {c x e +e}{-c d +e}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}+\frac {c^{2} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{4 e \left (c d +e \right )^{2}}-\frac {c^{2} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{4 e \left (c d -e \right )^{2}}-\frac {c^{2} b^{2} \arctanh \left (c x \right )^{2}}{2 \left (c x e +c d \right )^{2} e}-\frac {c^{2} b^{2} \ln \left (c x +1\right )^{2}}{8 e \left (c d -e \right )^{2}}-\frac {c^{2} b^{2} \ln \left (c x -1\right )^{2}}{8 e \left (c d +e \right )^{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 \[ -\frac {1}{2} \, {\left ({\left (\frac {4 \, c^{2} d \log \left (e x + d\right )}{c^{4} d^{4} - 2 \, c^{2} d^{2} e^{2} + e^{4}} - \frac {c \log \left (c x + 1\right )}{c^{2} d^{2} e - 2 \, c d e^{2} + e^{3}} + \frac {c \log \left (c x - 1\right )}{c^{2} d^{2} e + 2 \, c d e^{2} + e^{3}} - \frac {2}{c^{2} d^{3} - d e^{2} + {\left (c^{2} d^{2} e - e^{3}\right )} x}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right )} a b - \frac {1}{8} \, b^{2} {\left (\frac {\log \left (-c x + 1\right )^{2}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} + 2 \, \int -\frac {{\left (c e x - e\right )} \log \left (c x + 1\right )^{2} + {\left (c e x + c d - 2 \, {\left (c e x - e\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{c e^{4} x^{4} - d^{3} e + {\left (3 \, c d e^{3} - e^{4}\right )} x^{3} + 3 \, {\left (c d^{2} e^{2} - d e^{3}\right )} x^{2} + {\left (c d^{3} e - 3 \, d^{2} e^{2}\right )} x}\,{d x}\right )} - \frac {a^{2}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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