Integrand size = 16, antiderivative size = 125 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {i b \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )}{d^3}-\frac {i b \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )}{d^3} \]
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Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {14, 5544, 4265, 2611, 2320, 6724} \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}+\frac {i b \operatorname {PolyLog}\left (3,-i e^{d x^2+c}\right )}{d^3}-\frac {i b \operatorname {PolyLog}\left (3,i e^{d x^2+c}\right )}{d^3}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{d x^2+c}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{d x^2+c}\right )}{d^2} \]
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Rule 14
Rule 2320
Rule 2611
Rule 4265
Rule 5544
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^5+b x^5 \text {sech}\left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^6}{6}+b \int x^5 \text {sech}\left (c+d x^2\right ) \, dx \\ & = \frac {a x^6}{6}+\frac {1}{2} b \text {Subst}\left (\int x^2 \text {sech}(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {(i b) \text {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {(i b) \text {Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,x^2\right )}{d} \\ & = \frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {(i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {(i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2} \\ & = \frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3} \\ & = \frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {i b \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )}{d^3}-\frac {i b \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )}{d^3} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.14 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {i b \left (d^2 x^4 \log \left (1-i e^{c+d x^2}\right )-d^2 x^4 \log \left (1+i e^{c+d x^2}\right )-2 d x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )+2 d x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )+2 \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )-2 \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )\right )}{2 d^3} \]
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\[\int x^{5} \left (a +b \,\operatorname {sech}\left (d \,x^{2}+c \right )\right )d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (100) = 200\).
Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.05 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a d^{3} x^{6} + 6 i \, b d x^{2} {\rm Li}_2\left (i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right )\right ) - 6 i \, b d x^{2} {\rm Li}_2\left (-i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right )\right ) + 3 i \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + i\right ) - 3 i \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - i\right ) - 3 \, {\left (i \, b d^{2} x^{4} - i \, b c^{2}\right )} \log \left (i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right ) + 1\right ) - 3 \, {\left (-i \, b d^{2} x^{4} + i \, b c^{2}\right )} \log \left (-i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right ) + 1\right ) - 6 i \, b {\rm polylog}\left (3, i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right )\right ) + 6 i \, b {\rm polylog}\left (3, -i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right )\right )}{6 \, d^{3}} \]
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\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int x^{5} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )\, dx \]
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\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \]
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\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int x^5\,\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right ) \,d x \]
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