Optimal. Leaf size=546 \[ -\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^3}+\frac {a b^2 f^2 x}{d^2}-\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (c+d x) (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}+\frac {b^3 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_3\left (1-\frac {2}{-c-d x+1}\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3} \]
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Rubi [A] time = 1.05, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6111, 5928, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 6048, 6058, 6610} \[ -\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^3}+\frac {b^3 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^3}+\frac {a b^2 f^2 x}{d^2}-\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (c+d x) (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5928
Rule 5948
Rule 5980
Rule 5984
Rule 6048
Rule 6058
Rule 6111
Rule 6610
Rubi steps
\begin {align*} \int (e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \tanh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f) \left (a+b \tanh ^{-1}(x)\right )^2}{d^3}-\frac {f^3 x \left (a+b \tanh ^{-1}(x)\right )^2}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \operatorname {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b f^2\right ) \operatorname {Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}+\frac {(3 b f (d e-c f)) \operatorname {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \operatorname {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2}+\frac {f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2}\right ) \, dx,x,c+d x\right )}{d^3 f}-\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}+\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}-\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3 f}-\frac {\left (b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \operatorname {Subst}\left (\int \tanh ^{-1}(x) \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 b^3 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {\left (b^3 f^2\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^3 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d^3}+\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \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^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {\left (b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \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^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {Li}_3\left (1-\frac {2}{1-c-d x}\right )}{2 d^3}\\ \end {align*}
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Mathematica [B] time = 10.16, size = 1868, normalized size = 3.42 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} f^{2} x^{2} + 2 \, a^{3} e f x + a^{3} e^{2} + {\left (b^{3} f^{2} x^{2} + 2 \, b^{3} e f x + b^{3} e^{2}\right )} \operatorname {artanh}\left (d x + c\right )^{3} + 3 \, {\left (a b^{2} f^{2} x^{2} + 2 \, a b^{2} e f x + a b^{2} e^{2}\right )} \operatorname {artanh}\left (d x + c\right )^{2} + 3 \, {\left (a^{2} b f^{2} x^{2} + 2 \, a^{2} b e f x + a^{2} b e^{2}\right )} \operatorname {artanh}\left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 22.27, size = 12111, normalized size = 22.18 \[ \text {output too large to display} \]
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 \[ \text {result too large to display} \]
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 \[ \int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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