Integrand size = 12, antiderivative size = 111 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {x^6}{24 a}+\frac {1}{8} e^{\text {sech}^{-1}\left (a x^2\right )} x^8-\frac {x^2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{16 a^3}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \arcsin \left (a x^2\right )}{16 a^4} \]
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Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 265, 281, 327, 222} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \arcsin \left (a x^2\right )}{16 a^4}-\frac {x^2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{16 a^3}+\frac {x^6}{24 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )} \]
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Rule 30
Rule 222
Rule 265
Rule 281
Rule 327
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} e^{\text {sech}^{-1}\left (a x^2\right )} x^8+\frac {\int x^5 \, dx}{4 a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^5}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{4 a} \\ & = \frac {x^6}{24 a}+\frac {1}{8} e^{\text {sech}^{-1}\left (a x^2\right )} x^8+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^5}{\sqrt {1-a^2 x^4}} \, dx}{4 a} \\ & = \frac {x^6}{24 a}+\frac {1}{8} e^{\text {sech}^{-1}\left (a x^2\right )} x^8+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )}{8 a} \\ & = \frac {x^6}{24 a}+\frac {1}{8} e^{\text {sech}^{-1}\left (a x^2\right )} x^8-\frac {x^2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{16 a^3}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )}{16 a^3} \\ & = \frac {x^6}{24 a}+\frac {1}{8} e^{\text {sech}^{-1}\left (a x^2\right )} x^8-\frac {x^2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{16 a^3}+\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \arcsin \left (a x^2\right )}{16 a^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {8 a^3 x^6-3 a \sqrt {\frac {1-a x^2}{1+a x^2}} \left (x^2+a x^4-2 a^2 x^6-2 a^3 x^8\right )+3 i \log \left (-2 i a x^2+2 \sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )\right )}{48 a^4} \]
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Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (2 x^{6} \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}\, a^{4}-x^{2} \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}\, a^{2}+\arctan \left (\frac {x^{2}}{\sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}}\right )\right )}{16 \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}\, a^{4}}+\frac {x^{6}}{6 a}\) | \(137\) |
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none
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {8 \, a^{3} x^{6} + 3 \, {\left (2 \, a^{4} x^{8} - a^{2} x^{4}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 6 \, \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right )}{48 \, a^{4}} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {\int x^{5}\, dx + \int a x^{7} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\int { x^{7} {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (76) = 152\).
Time = 0.32 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.85 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {8 \, a^{2} x^{6} + 4 \, \sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (a^{2} x^{2} + a\right )} {\left (\frac {2 \, {\left (a^{2} x^{2} + a\right )}}{a^{4}} - \frac {7}{a^{3}}\right )} + \frac {9}{a^{2}}\right )} + {\left (\sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (a^{2} x^{2} + a\right )} {\left (2 \, {\left (a^{2} x^{2} + a\right )} {\left (\frac {3 \, {\left (a^{2} x^{2} + a\right )}}{a^{6}} - \frac {13}{a^{5}}\right )} + \frac {43}{a^{4}}\right )} - \frac {39}{a^{3}}\right )} - \frac {18 \, \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right )}{a^{2}}\right )} a + \frac {24 \, \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right )}{a}}{48 \, a^{3}} \]
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Time = 18.82 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.69 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{16\,a^4}-\frac {\frac {1{}\mathrm {i}}{2048\,a^4}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4\,11{}\mathrm {i}}{1024\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^6\,7{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^6}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^8\,239{}\mathrm {i}}{2048\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^{10}\,1{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^{12}}}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{16\,a^4}+\frac {x^6}{6\,a}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2048\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4} \]
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