Optimal. Leaf size=53 \[ \frac {x^2 \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\tanh ^{-1}(\tanh (a+b x))^6}{60 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2188, 30}
\begin {gather*} \frac {\tanh ^{-1}(\tanh (a+b x))^6}{60 b^3}-\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^4}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rule 2199
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {\int x \tanh ^{-1}(\tanh (a+b x))^4 \, dx}{2 b}\\ &=\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\int \tanh ^{-1}(\tanh (a+b x))^5 \, dx}{10 b^2}\\ &=\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\text {Subst}\left (\int x^5 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{10 b^3}\\ &=\frac {x^2 \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{10 b^2}+\frac {\tanh ^{-1}(\tanh (a+b x))^6}{60 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 1.02 \begin {gather*} -\frac {1}{60} x^3 \left (b^3 x^3-6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+15 b x \tanh ^{-1}(\tanh (a+b x))^2-20 \tanh ^{-1}(\tanh (a+b x))^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 56, normalized size = 1.06 \[\frac {x^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3}-b \left (\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{4}-\frac {b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}{5}-\frac {b \,x^{6}}{30}\right )}{2}\right )\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 54, normalized size = 1.02 \begin {gather*} -\frac {1}{4} \, b x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac {1}{3} \, x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac {1}{60} \, {\left (b^{2} x^{6} - 6 \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 35, normalized size = 0.66 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 56, normalized size = 1.06 \begin {gather*} - \frac {b^{3} x^{6}}{60} + \frac {b^{2} x^{5} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{10} - \frac {b x^{4} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4} + \frac {x^{3} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 35, normalized size = 0.66 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 53, normalized size = 1.00 \begin {gather*} -\frac {b^3\,x^6}{60}+\frac {b^2\,x^5\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{10}-\frac {b\,x^4\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4}+\frac {x^3\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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