Optimal. Leaf size=71 \[ \frac {2 (a+1)^4 \log (a+b x+1)}{b^5}-\frac {2 (a+1)^3 x}{b^4}+\frac {(a+1)^2 x^2}{b^3}-\frac {2 (a+1) x^3}{3 b^2}+\frac {x^4}{2 b}-\frac {x^5}{5} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6163, 77} \[ -\frac {2 (a+1) x^3}{3 b^2}+\frac {(a+1)^2 x^2}{b^3}-\frac {2 (a+1)^3 x}{b^4}+\frac {2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac {x^4}{2 b}-\frac {x^5}{5} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6163
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a+b x)} x^4 \, dx &=\int \frac {x^4 (1-a-b x)}{1+a+b x} \, dx\\ &=\int \left (-\frac {2 (1+a)^3}{b^4}+\frac {2 (1+a)^2 x}{b^3}-\frac {2 (1+a) x^2}{b^2}+\frac {2 x^3}{b}-x^4+\frac {2 (1+a)^4}{b^4 (1+a+b x)}\right ) \, dx\\ &=-\frac {2 (1+a)^3 x}{b^4}+\frac {(1+a)^2 x^2}{b^3}-\frac {2 (1+a) x^3}{3 b^2}+\frac {x^4}{2 b}-\frac {x^5}{5}+\frac {2 (1+a)^4 \log (1+a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 71, normalized size = 1.00 \[ \frac {2 (a+1)^4 \log (a+b x+1)}{b^5}-\frac {2 (a+1)^3 x}{b^4}+\frac {(a+1)^2 x^2}{b^3}-\frac {2 (a+1) x^3}{3 b^2}+\frac {x^4}{2 b}-\frac {x^5}{5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 93, normalized size = 1.31 \[ -\frac {6 \, b^{5} x^{5} - 15 \, b^{4} x^{4} + 20 \, {\left (a + 1\right )} b^{3} x^{3} - 30 \, {\left (a^{2} + 2 \, a + 1\right )} b^{2} x^{2} + 60 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} b x - 60 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{30 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 202, normalized size = 2.85 \[ \frac {{\left (b x + a + 1\right )}^{5} {\left (\frac {15 \, {\left (2 \, a b + 3 \, b\right )}}{{\left (b x + a + 1\right )} b} - \frac {20 \, {\left (3 \, a^{2} b^{2} + 10 \, a b^{2} + 7 \, b^{2}\right )}}{{\left (b x + a + 1\right )}^{2} b^{2}} + \frac {60 \, {\left (a^{3} b^{3} + 6 \, a^{2} b^{3} + 9 \, a b^{3} + 4 \, b^{3}\right )}}{{\left (b x + a + 1\right )}^{3} b^{3}} - \frac {30 \, {\left (a^{4} b^{4} + 12 \, a^{3} b^{4} + 30 \, a^{2} b^{4} + 28 \, a b^{4} + 9 \, b^{4}\right )}}{{\left (b x + a + 1\right )}^{4} b^{4}} - 6\right )}}{30 \, b^{5}} - \frac {2 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2} {\left | b \right |}}\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 159, normalized size = 2.24 \[ -\frac {x^{5}}{5}+\frac {x^{4}}{2 b}-\frac {2 x^{3} a}{3 b^{2}}-\frac {2 x^{3}}{3 b^{2}}+\frac {x^{2} a^{2}}{b^{3}}+\frac {2 x^{2} a}{b^{3}}-\frac {2 x \,a^{3}}{b^{4}}+\frac {x^{2}}{b^{3}}-\frac {6 a^{2} x}{b^{4}}-\frac {6 a x}{b^{4}}-\frac {2 x}{b^{4}}+\frac {2 \ln \left (b x +a +1\right ) a^{4}}{b^{5}}+\frac {8 \ln \left (b x +a +1\right ) a^{3}}{b^{5}}+\frac {12 \ln \left (b x +a +1\right ) a^{2}}{b^{5}}+\frac {8 \ln \left (b x +a +1\right ) a}{b^{5}}+\frac {2 \ln \left (b x +a +1\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 94, normalized size = 1.32 \[ -\frac {6 \, b^{4} x^{5} - 15 \, b^{3} x^{4} + 20 \, {\left (a + 1\right )} b^{2} x^{3} - 30 \, {\left (a^{2} + 2 \, a + 1\right )} b x^{2} + 60 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} x}{30 \, b^{4}} + \frac {2 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 141, normalized size = 1.99 \[ \frac {\ln \left (a+b\,x+1\right )\,\left (2\,a^4+8\,a^3+12\,a^2+8\,a+2\right )}{b^5}-\frac {x^5}{5}-x^4\,\left (\frac {a-1}{4\,b}-\frac {a+1}{4\,b}\right )-\frac {x^2\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )\,{\left (a+1\right )}^2}{2\,b^2}+\frac {x^3\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )\,\left (a+1\right )}{3\,b}+\frac {x\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )\,{\left (a+1\right )}^3}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 102, normalized size = 1.44 \[ - \frac {x^{5}}{5} - x^{3} \left (\frac {2 a}{3 b^{2}} + \frac {2}{3 b^{2}}\right ) - x^{2} \left (- \frac {a^{2}}{b^{3}} - \frac {2 a}{b^{3}} - \frac {1}{b^{3}}\right ) - x \left (\frac {2 a^{3}}{b^{4}} + \frac {6 a^{2}}{b^{4}} + \frac {6 a}{b^{4}} + \frac {2}{b^{4}}\right ) + \frac {x^{4}}{2 b} + \frac {2 \left (a + 1\right )^{4} \log {\left (a + b x + 1 \right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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