Optimal. Leaf size=359 \[ \frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {2 b d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}-\frac {b^2 d \left (c^2 d^2+e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3} \]
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Rubi [A]
time = 0.38, antiderivative size = 359, normalized size of antiderivative = 1.00, number
of steps used = 19, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules
used = {6065, 6021, 266, 6037, 327, 212, 272, 45, 6195, 6095, 6131, 6055, 2449, 2352}
\begin {gather*} -\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {a b e x \left (6 c^2 d^2+e^2\right )}{2 c^3}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {2 b d \left (c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}-\frac {b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac {b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2}{12 c^2}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{12 c^4}-\frac {b^2 d \left (c^2 d^2+e^2\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 e x \left (6 c^2 d^2+e^2\right ) \tanh ^{-1}(c x)}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 266
Rule 272
Rule 327
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6065
Rule 6095
Rule 6131
Rule 6195
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \left (-\frac {e^2 \left (6 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^4}-\frac {4 d e^3 x \left (a+b \tanh ^{-1}(c x)\right )}{c^2}-\frac {e^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4+4 c^2 d e \left (c^2 d^2+e^2\right ) x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {b \int \frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4+4 c^2 d e \left (c^2 d^2+e^2\right ) x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{2 c^3 e}+\frac {\left (2 b d e^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}+\frac {\left (b e^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}+\frac {\left (b e \left (6 c^2 d^2+e^2\right )\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^3}\\ &=\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {b \int \left (\frac {c^4 d^4 \left (1+\frac {6 c^2 d^2 e^2+e^4}{c^4 d^4}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}+\frac {4 c^2 d e \left (c^2 d^2+e^2\right ) x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{2 c^3 e}-\left (b^2 d e^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{6} \left (b^2 e^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx+\frac {\left (b^2 e \left (6 c^2 d^2+e^2\right )\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^3}\\ &=\frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {\left (b^2 d e^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c^2}-\frac {1}{12} \left (b^2 e^3\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {\left (2 b d \left (c^2 d^2+e^2\right )\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c}-\frac {\left (b^2 e \left (6 c^2 d^2+e^2\right )\right ) \int \frac {x}{1-c^2 x^2} \, dx}{2 c^2}-\frac {\left (b \left (c^4 d^4+6 c^2 d^2 e^2+e^4\right )\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^3 e}\\ &=\frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}-\frac {b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}-\frac {1}{12} \left (b^2 e^3\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\left (2 b d \left (c^2 d^2+e^2\right )\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^2}\\ &=\frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {2 b d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {\left (2 b^2 d \left (c^2 d^2+e^2\right )\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {2 b d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}-\frac {\left (2 b^2 d \left (c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^3}\\ &=\frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tanh ^{-1}(c x)}{c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {b e^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 e}-\frac {2 b d \left (c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}-\frac {b^2 d \left (c^2 d^2+e^2\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 506, normalized size = 1.41 \begin {gather*} \frac {-b^2 e^3+12 a^2 c^4 d^3 x+36 a b c^3 d^2 e x+12 b^2 c^2 d e^2 x+6 a b c e^3 x+18 a^2 c^4 d^2 e x^2+12 a b c^3 d e^2 x^2+b^2 c^2 e^3 x^2+12 a^2 c^4 d e^2 x^3+2 a b c^3 e^3 x^3+3 a^2 c^4 e^3 x^4+3 b^2 \left (-4 c^3 d^3-6 c^2 d^2 e-4 c d e^2-e^3+c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \tanh ^{-1}(c x)^2+2 b c \tanh ^{-1}(c x) \left (3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b e \left (18 c^2 d^2 x+6 d e \left (-1+c^2 x^2\right )+e^2 x \left (3+c^2 x^2\right )\right )-12 b d \left (c^2 d^2+e^2\right ) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+18 a b c^2 d^2 e \log (1-c x)+3 a b e^3 \log (1-c x)-18 a b c^2 d^2 e \log (1+c x)-3 a b e^3 \log (1+c x)+12 a b c^3 d^3 \log \left (1-c^2 x^2\right )+18 b^2 c^2 d^2 e \log \left (1-c^2 x^2\right )+4 b^2 e^3 \log \left (1-c^2 x^2\right )+12 a b c d e^2 \log \left (-1+c^2 x^2\right )+12 b^2 c d \left (c^2 d^2+e^2\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{12 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1372\) vs.
\(2(343)=686\).
time = 0.38, size = 1373, normalized size = 3.82
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1373\) |
default | \(\text {Expression too large to display}\) | \(1373\) |
risch | \(\text {Expression too large to display}\) | \(1560\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 808 vs.
\(2 (328) = 656\).
time = 0.43, size = 808, normalized size = 2.25 \begin {gather*} \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x + \frac {3}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b d^{2} e + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{3}}{c} + {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b d e^{2} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a b e^{3} + \frac {{\left (b^{2} c^{2} d^{3} + b^{2} d e^{2}\right )} {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )}}{c^{3}} + \frac {{\left (9 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{2} e^{3}\right )} \log \left (c x + 1\right )}{6 \, c^{4}} + \frac {{\left (9 \, b^{2} c^{2} d^{2} e + 3 \, b^{2} c d e^{2} + 2 \, b^{2} e^{3}\right )} \log \left (c x - 1\right )}{6 \, c^{4}} + \frac {4 \, b^{2} c^{2} x^{2} e^{3} + 48 \, b^{2} c^{2} d x e^{2} + 3 \, {\left (b^{2} c^{4} x^{4} e^{3} + 4 \, b^{2} c^{4} d x^{3} e^{2} + 6 \, b^{2} c^{4} d^{2} x^{2} e + 4 \, b^{2} c^{4} d^{3} x + 4 \, b^{2} c^{3} d^{3} - 6 \, b^{2} c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{2} e^{3}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (b^{2} c^{4} x^{4} e^{3} + 4 \, b^{2} c^{4} d x^{3} e^{2} + 6 \, b^{2} c^{4} d^{2} x^{2} e + 4 \, b^{2} c^{4} d^{3} x - 4 \, b^{2} c^{3} d^{3} - 6 \, b^{2} c^{2} d^{2} e - 4 \, b^{2} c d e^{2} - b^{2} e^{3}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (b^{2} c^{3} x^{3} e^{3} + 6 \, b^{2} c^{3} d x^{2} e^{2} + 3 \, {\left (6 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (2 \, b^{2} c^{3} x^{3} e^{3} + 12 \, b^{2} c^{3} d x^{2} e^{2} + 6 \, {\left (6 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x + 3 \, {\left (b^{2} c^{4} x^{4} e^{3} + 4 \, b^{2} c^{4} d x^{3} e^{2} + 6 \, b^{2} c^{4} d^{2} x^{2} e + 4 \, b^{2} c^{4} d^{3} x + 4 \, b^{2} c^{3} d^{3} - 6 \, b^{2} c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{2} e^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{48 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{3}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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