Optimal. Leaf size=269 \[ -\frac {b^2 \text {Li}_2\left (\frac {2}{c+d x+1}-1\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4}-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}+\frac {b \log \left (2-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^4}-\frac {b^3 \text {Li}_3\left (\frac {2}{c+d x+1}-1\right )}{2 d e^4}+\frac {b^3 \log (c+d x)}{d e^4}-\frac {b^3 \log \left (1-(c+d x)^2\right )}{2 d e^4} \]
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Rubi [A] time = 0.51, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6107, 12, 5916, 5982, 266, 36, 31, 29, 5948, 5988, 5932, 6056, 6610} \[ -\frac {b^2 \text {PolyLog}\left (2,\frac {2}{c+d x+1}-1\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4}-\frac {b^3 \text {PolyLog}\left (3,\frac {2}{c+d x+1}-1\right )}{2 d e^4}-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}+\frac {b \log \left (2-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d e^4}+\frac {b^3 \log (c+d x)}{d e^4}-\frac {b^3 \log \left (1-(c+d x)^2\right )}{2 d e^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 266
Rule 5916
Rule 5932
Rule 5948
Rule 5982
Rule 5988
Rule 6056
Rule 6107
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{(c e+d e x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^3}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^3}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x^3 \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x^3} \, dx,x,c+d x\right )}{d e^4}+\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x (1+x)} \, dx,x,c+d x\right )}{d e^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^2 \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (2-\frac {2}{1+c+d x}\right )}{d e^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{d e^4}-\frac {\left (2 b^2\right ) \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^4}\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (2-\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c+d x}\right )}{d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^4}+\frac {b^3 \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^4}\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (2-\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^3 \text {Li}_3\left (-1+\frac {2}{1+c+d x}\right )}{2 d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,(c+d x)^2\right )}{2 d e^4}\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (2-\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^3 \text {Li}_3\left (-1+\frac {2}{1+c+d x}\right )}{2 d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,(c+d x)^2\right )}{2 d e^4}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,(c+d x)^2\right )}{2 d e^4}\\ &=-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b^3 \log (c+d x)}{d e^4}-\frac {b^3 \log \left (1-(c+d x)^2\right )}{2 d e^4}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (2-\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^3 \text {Li}_3\left (-1+\frac {2}{1+c+d x}\right )}{2 d e^4}\\ \end {align*}
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Mathematica [C] time = 1.33, size = 393, normalized size = 1.46 \[ \frac {-\frac {2 a^3}{(c+d x)^3}-3 a^2 b \log \left (-c^2-2 c d x-d^2 x^2+1\right )-\frac {3 a^2 b}{(c+d x)^2}+6 a^2 b \log (c+d x)-\frac {6 a^2 b \tanh ^{-1}(c+d x)}{(c+d x)^3}+6 a b^2 \left (-\text {Li}_2\left (e^{-2 \tanh ^{-1}(c+d x)}\right )-\frac {(c+d x)^2+\tanh ^{-1}(c+d x)^2}{(c+d x)^3}+\tanh ^{-1}(c+d x) \left (-\frac {1-(c+d x)^2}{(c+d x)^2}+\tanh ^{-1}(c+d x)+2 \log \left (1-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )\right )+6 b^3 \left (\tanh ^{-1}(c+d x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c+d x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \tanh ^{-1}(c+d x)}\right )+\log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )-\frac {\left (1-(c+d x)^2\right ) \tanh ^{-1}(c+d x)^3}{3 (c+d x)^3}-\frac {\tanh ^{-1}(c+d x)^3}{3 (c+d x)}-\frac {1}{3} \tanh ^{-1}(c+d x)^3-\frac {\left (1-(c+d x)^2\right ) \tanh ^{-1}(c+d x)^2}{2 (c+d x)^2}-\frac {\tanh ^{-1}(c+d x)}{c+d x}+\tanh ^{-1}(c+d x)^2 \log \left (1-e^{2 \tanh ^{-1}(c+d x)}\right )+\frac {i \pi ^3}{24}\right )}{6 d e^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (d x + c\right ) + a^{3}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.98, size = 2247, normalized size = 8.35 \[ \text {Expression too large to display} \]
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 (d {\left (\frac {1}{d^{4} e^{4} x^{2} + 2 \, c d^{3} e^{4} x + c^{2} d^{2} e^{4}} + \frac {\log \left (d x + c + 1\right )}{d^{2} e^{4}} - \frac {2 \, \log \left (d x + c\right )}{d^{2} e^{4}} + \frac {\log \left (d x + c - 1\right )}{d^{2} e^{4}}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}}\right )} a^{2} b - \frac {a^{3}}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} - \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + {\left (c^{3} - 1\right )} b^{3}\right )} \log \left (-d x - c + 1\right )^{3} + 3 \, {\left (b^{3} d x + b^{3} c + 2 \, a b^{2} + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + {\left (c^{3} + 1\right )} b^{3}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )^{2}}{24 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} - \int -\frac {{\left (b^{3} d x + b^{3} {\left (c - 1\right )}\right )} \log \left (d x + c + 1\right )^{3} + 6 \, {\left (a b^{2} d x + a b^{2} {\left (c - 1\right )}\right )} \log \left (d x + c + 1\right )^{2} + {\left (2 \, b^{3} d^{2} x^{2} + 2 \, b^{3} c^{2} + 4 \, a b^{2} c - 3 \, {\left (b^{3} d x + b^{3} {\left (c - 1\right )}\right )} \log \left (d x + c + 1\right )^{2} + 4 \, {\left (b^{3} c d + a b^{2} d\right )} x + 2 \, {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + {\left (c^{4} + c\right )} b^{3} - 6 \, a b^{2} {\left (c - 1\right )} + {\left ({\left (4 \, c^{3} d + d\right )} b^{3} - 6 \, a b^{2} d\right )} x\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{8 \, {\left (d^{5} e^{4} x^{5} + c^{5} e^{4} - c^{4} e^{4} + {\left (5 \, c d^{4} e^{4} - d^{4} e^{4}\right )} x^{4} + 2 \, {\left (5 \, c^{2} d^{3} e^{4} - 2 \, c d^{3} e^{4}\right )} x^{3} + 2 \, {\left (5 \, c^{3} d^{2} e^{4} - 3 \, c^{2} d^{2} e^{4}\right )} x^{2} + {\left (5 \, c^{4} d e^{4} - 4 \, c^{3} d e^{4}\right )} x\right )}}\,{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 = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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 \[ \frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {atanh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {atanh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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