Optimal. Leaf size=147 \[ -\frac {x}{a^4}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5} \]
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Rubi [A]
time = 0.40, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6472, 1818,
1828, 12, 267} \begin {gather*} -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^5}{5 a^5}+\frac {\left (16 \sqrt {\frac {1-a x}{a x+1}}+5\right ) (a x+1)^4}{20 a^5}-\frac {\left (17 \sqrt {\frac {1-a x}{a x+1}}+15\right ) (a x+1)^3}{15 a^5}+\frac {\left (4 \sqrt {\frac {1-a x}{a x+1}}+9\right ) (a x+1)^2}{6 a^5}-\frac {x}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 267
Rule 1818
Rule 1828
Rule 6472
Rubi steps
\begin {align*} \int e^{-\text {sech}^{-1}(a x)} x^4 \, dx &=\int \frac {x^4}{\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}} \, dx\\ &=\frac {4 \text {Subst}\left (\int \frac {(-1+x)^5 x (1+x)^3}{\left (1+x^2\right )^6} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}-\frac {2 \text {Subst}\left (\int \frac {-16+10 x+140 x^2-30 x^3-80 x^4+30 x^5+20 x^6-10 x^7}{\left (1+x^2\right )^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{5 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}+\frac {\text {Subst}\left (\int \frac {-128+560 x+800 x^2-320 x^3-160 x^4+80 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}-\frac {\text {Subst}\left (\int \frac {-320+2400 x+960 x^2-480 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{120 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}+\frac {\text {Subst}\left (\int \frac {1920 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{480 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}+\frac {4 \text {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^5}\\ &=-\frac {x}{a^4}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 65, normalized size = 0.44 \begin {gather*} \frac {15 a^4 x^4-4 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-2+2 a x-3 a^2 x^2+3 a^3 x^3\right )}{60 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.22, size = 531, normalized size = 3.61
method | result | size |
default | \(\frac {\left (a x +1\right ) \left (15 a^{10} x^{10} \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}}+30 a^{8} x^{8} \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}}-30 \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {3}{2}} a^{8} x^{8}+30 x^{6} \ln \left (a^{2} x^{2}\right ) \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}} a^{6}-30 a^{7} x^{7} \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {3}{2}}-60 \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {3}{2}} \ln \left (a^{2} x^{2}\right ) a^{6} x^{6}-12 a^{11} x^{11}-60 x^{5} \ln \left (a^{2} x^{2}\right ) \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {3}{2}} a^{5}+30 \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \sqrt {\frac {a x +1}{a x}}\, \ln \left (a^{2} x^{2}\right ) a^{6} x^{6}+12 a^{10} x^{10}+60 \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \sqrt {\frac {a x +1}{a x}}\, \ln \left (a^{2} x^{2}\right ) a^{5} x^{5}+40 a^{9} x^{9}+30 x^{4} \ln \left (a^{2} x^{2}\right ) \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \sqrt {\frac {a x +1}{a x}}\, a^{4}-40 a^{8} x^{8}-40 a^{7} x^{7}+40 a^{6} x^{6}+20 a^{3} x^{3}-20 a^{2} x^{2}-8 a x +8\right )}{60 x^{7} a^{12} \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}}}\) | \(531\) |
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 [A]
time = 0.41, size = 65, normalized size = 0.44 \begin {gather*} \frac {15 \, a^{3} x^{4} - 4 \, {\left (3 \, a^{4} x^{5} - a^{2} x^{3} - 2 \, x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{60 \, a^{4}} \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*} a \int \frac {x^{5}}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, 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 [B]
time = 2.14, size = 73, normalized size = 0.50 \begin {gather*} \frac {x^4}{4\,a}+\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,x}{15\,a^4}+\frac {2}{15\,a^5}-\frac {x^5}{5}-\frac {x^4}{5\,a}+\frac {x^3}{15\,a^2}+\frac {x^2}{15\,a^3}\right )}{\sqrt {\frac {1}{a\,x}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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