Optimal. Leaf size=270 \[ \frac {a b d x}{2 c^3}+\frac {3 b^2 d x}{10 c^3}+\frac {b^2 d x^2}{12 c^2}+\frac {b^2 d x^3}{30 c}-\frac {3 b^2 d \tanh ^{-1}(c x)}{10 c^4}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^4}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^4}-\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^4} \]
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Rubi [A]
time = 0.46, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps
used = 27, number of rules used = 15, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used =
{6087, 6037, 6127, 272, 45, 6021, 266, 6095, 308, 212, 327, 6131, 6055, 2449, 2352}
\begin {gather*} -\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^4}+\frac {a b d x}{2 c^3}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^4}-\frac {3 b^2 d \tanh ^{-1}(c x)}{10 c^4}+\frac {3 b^2 d x}{10 c^3}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^3}+\frac {b^2 d x^2}{12 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^4}+\frac {b^2 d x^3}{30 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 266
Rule 272
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6087
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x^3 (d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+(c d) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{2} (b c d) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{5} \left (2 b c^2 d\right ) \int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} (2 b d) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{5} (2 b d) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {(b d) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}-\frac {(b d) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{2 c}\\ &=\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{6} \left (b^2 d\right ) \int \frac {x^3}{1-c^2 x^2} \, dx+\frac {(b d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac {(b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^3}+\frac {(2 b d) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac {(2 b d) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^2}-\frac {1}{10} \left (b^2 c d\right ) \int \frac {x^4}{1-c^2 x^2} \, dx\\ &=\frac {a b d x}{2 c^3}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{12} \left (b^2 d\right ) \text {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {(2 b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^3}+\frac {\left (b^2 d\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^3}-\frac {\left (b^2 d\right ) \int \frac {x^2}{1-c^2 x^2} \, dx}{5 c}-\frac {1}{10} \left (b^2 c d\right ) \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {a b d x}{2 c^3}+\frac {3 b^2 d x}{10 c^3}+\frac {b^2 d x^3}{30 c}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^4}-\frac {1}{12} \left (b^2 d\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 (b^2 d\right ) \int \frac {1}{1-c^2 x^2} \, dx}{10 c^3}-\frac {\left (b^2 d\right ) \int \frac {1}{1-c^2 x^2} \, dx}{5 c^3}+\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^3}-\frac {\left (b^2 d\right ) \int \frac {x}{1-c^2 x^2} \, dx}{2 c^2}\\ &=\frac {a b d x}{2 c^3}+\frac {3 b^2 d x}{10 c^3}+\frac {b^2 d x^2}{12 c^2}+\frac {b^2 d x^3}{30 c}-\frac {3 b^2 d \tanh ^{-1}(c x)}{10 c^4}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^4}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^4}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^4}\\ &=\frac {a b d x}{2 c^3}+\frac {3 b^2 d x}{10 c^3}+\frac {b^2 d x^2}{12 c^2}+\frac {b^2 d x^3}{30 c}-\frac {3 b^2 d \tanh ^{-1}(c x)}{10 c^4}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^3}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac {1}{10} b d x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^4}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^4}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 271, normalized size = 1.00 \begin {gather*} \frac {d \left (-18 a b-5 b^2+30 a b c x+18 b^2 c x+12 a b c^2 x^2+5 b^2 c^2 x^2+10 a b c^3 x^3+2 b^2 c^3 x^3+15 a^2 c^4 x^4+6 a b c^4 x^4+12 a^2 c^5 x^5+3 b^2 \left (-9+5 c^4 x^4+4 c^5 x^5\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (3 a c^4 x^4 (5+4 c x)+b \left (-9+15 c x+6 c^2 x^2+5 c^3 x^3+3 c^4 x^4\right )-12 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+15 a b \log (1-c x)-15 a b \log (1+c x)+20 b^2 \log \left (1-c^2 x^2\right )+12 a b \log \left (-1+c^2 x^2\right )+12 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{60 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 403, normalized size = 1.49 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 \begin {gather*} \text {Failed to integrate} \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*} d \left (\int a^{2} x^{3}\, dx + \int a^{2} c x^{4}\, dx + \int b^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int b^{2} c x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x^{4} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \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 x^3\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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