Optimal. Leaf size=203 \[ \frac {(1-a x) (a x+1)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)^4}{10 a^5}-\frac {\left (45 \sqrt {\frac {1-a x}{a x+1}}+4\right ) (a x+1)^3}{30 a^5}+\frac {5 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2}{4 a^5}+\frac {\left (4-\sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)}{4 a^5}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{2 a^5} \]
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Rubi [A] time = 0.70, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6337, 1804, 1811, 1814, 639, 203} \[ \frac {(1-a x) (a x+1)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{a x+1}} \left (5-6 \sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)^4}{10 a^5}-\frac {\left (45 \sqrt {\frac {1-a x}{a x+1}}+4\right ) (a x+1)^3}{30 a^5}+\frac {5 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2}{4 a^5}+\frac {\left (4-\sqrt {\frac {1-a x}{a x+1}}\right ) (a x+1)}{4 a^5}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{2 a^5} \]
Antiderivative was successfully verified.
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Rule 203
Rule 639
Rule 1804
Rule 1811
Rule 1814
Rule 6337
Rubi steps
\begin {align*} \int e^{2 \text {sech}^{-1}(a x)} x^4 \, dx &=\int x^4 \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2 \, dx\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {(-1+x)^2 x (1+x)^6}{\left (1+x^2\right )^6} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^5}\\ &=\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {-42 x-40 x^2+130 x^3+80 x^4-30 x^5-40 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 {(1-a x) (1+a x)^4}{5 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-42-40 x+130 x^2+80 x^3-30 x^4-40 x^5-10 x^6\right )}{\left (1+x^2\right )^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{5 a^5}\\ &=\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}-\frac {\operatorname {Subst}\left (\int \frac {160-48 x-960 x^2+160 x^3+320 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 {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}+\frac {\operatorname {Subst}\left (\int \frac {480-480 x-1920 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 {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\operatorname {Subst}\left (\int \frac {480+1920 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{480 a^5}\\ &=\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}+\frac {(1+a x) \left (4-\sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^5}\\ &=\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{4 a^5}+\frac {(1-a x) (1+a x)^4}{5 a^5}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4 \left (5-6 \sqrt {\frac {1-a x}{1+a x}}\right )}{10 a^5}+\frac {(1+a x) \left (4-\sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^5}-\frac {(1+a x)^3 \left (4+45 \sqrt {\frac {1-a x}{1+a x}}\right )}{30 a^5}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )}{2 a^5}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 105, normalized size = 0.52 \[ \frac {-12 a^5 x^5+40 a^3 x^3-15 a \sqrt {\frac {1-a x}{a x+1}} \left (-2 a^3 x^4-2 a^2 x^3+a x^2+x\right )+15 i \log \left (2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)-2 i a x\right )}{60 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.89, size = 103, normalized size = 0.51 \[ -\frac {12 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 15 \, {\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 15 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{60 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 123, normalized size = 0.61 \[ -\frac {x^{5}}{5}+\frac {2 x^{3}}{3 a^{2}}+\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (2 \,\mathrm {csgn}\relax (a ) x^{3} a^{3} \sqrt {-a^{2} x^{2}+1}-x \sqrt {-a^{2} x^{2}+1}\, \mathrm {csgn}\relax (a ) a +\arctan \left (\frac {\mathrm {csgn}\relax (a ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \mathrm {csgn}\relax (a )}{4 a^{4} \sqrt {-a^{2} x^{2}+1}} \]
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 \[ \frac {2 \, x^{3}}{3 \, a^{2}} + \frac {2 \, {\left (-\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{2}} + \frac {\arcsin \left (a x\right )}{8 \, a^{3}}\right )}}{a^{2}} - \int x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 19.73, size = 808, normalized size = 3.98 \[ -\frac {\frac {1{}\mathrm {i}}{512\,a^5}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,3{}\mathrm {i}}{64\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,53{}\mathrm {i}}{256\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,87{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,657{}\mathrm {i}}{512\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}\,121{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}}-\frac {\frac {1{}\mathrm {i}}{16\,a^5}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{8\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{16\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}}-\frac {x^5\,\left (\frac {a^2}{5}-\frac {2}{3\,x^2}\right )}{a^2}-\frac {\ln \left (\frac {a\,\sqrt {\frac {1}{a\,x}+1}-\frac {1}{x}+a\,\sqrt {\frac {1}{a\,x}-1}\,1{}\mathrm {i}}{2\,a-2\,a\,\sqrt {\frac {1}{a\,x}+1}+\frac {1}{x}}\right )\,3{}\mathrm {i}}{4\,a^5}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{4\,a^5}+\frac {\ln \left (\frac {2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}-\frac {2}{x}+a\,\sqrt {-\frac {a-\frac {1}{x}}{a}}\,2{}\mathrm {i}}{2\,a+\frac {1}{x}-2\,a\,\sqrt {\frac {a+\frac {1}{x}}{a}}}\right )\,1{}\mathrm {i}}{a^5}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{128\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{512\,a^5\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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