Integrand size = 18, antiderivative size = 217 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \log \left (1+e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {2 i a b \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )}{d^3}-\frac {2 i a b \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )}{d^3}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d} \]
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Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5544, 4275, 4265, 2611, 2320, 6724, 4269, 3799, 2221, 2317, 2438} \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}+\frac {2 i a b \operatorname {PolyLog}\left (3,-i e^{d x^2+c}\right )}{d^3}-\frac {2 i a b \operatorname {PolyLog}\left (3,i e^{d x^2+c}\right )}{d^3}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{d x^2+c}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{d x^2+c}\right )}{d^2}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \left (d x^2+c\right )}\right )}{2 d^3}-\frac {b^2 x^2 \log \left (e^{2 \left (c+d x^2\right )}+1\right )}{d^2}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d}+\frac {b^2 x^4}{2 d} \]
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3799
Rule 4265
Rule 4269
Rule 4275
Rule 5544
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (a+b \text {sech}(c+d x))^2 \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text {sech}(c+d x)+b^2 x^2 \text {sech}^2(c+d x)\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^6}{6}+(a b) \text {Subst}\left (\int x^2 \text {sech}(c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \text {Subst}\left (\int x^2 \text {sech}^2(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d}-\frac {(2 i a b) \text {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {(2 i a b) \text {Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,x^2\right )}{d}-\frac {b^2 \text {Subst}\left (\int x \tanh (c+d x) \, dx,x,x^2\right )}{d} \\ & = \frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d}+\frac {(2 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {(2 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x}{1+e^{2 (c+d x)}} \, dx,x,x^2\right )}{d} \\ & = \frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \log \left (1+e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d}+\frac {(2 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {(2 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}+\frac {b^2 \text {Subst}\left (\int \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,x^2\right )}{d^2} \\ & = \frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \log \left (1+e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {2 i a b \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )}{d^3}-\frac {2 i a b \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )}{d^3}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d}+\frac {b^2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (c+d x^2\right )}\right )}{2 d^3} \\ & = \frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}+\frac {2 a b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \log \left (1+e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {2 i a b \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )}{d^3}-\frac {2 i a b \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )}{d^3}+\frac {b^2 x^4 \tanh \left (c+d x^2\right )}{2 d} \\ \end{align*}
Time = 2.71 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.47 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {\cosh \left (c+d x^2\right ) \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \left (a^2 x^6 \cosh \left (c+d x^2\right )+\frac {3 b \cosh \left (c+d x^2\right ) \left (2 b d^2 e^{2 c} x^4-2 b d^2 \left (1+e^{2 c}\right ) x^4+b \left (1+e^{2 c}\right ) \left (2 d x^2 \left (d x^2-\log \left (1+e^{2 \left (c+d x^2\right )}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 \left (c+d x^2\right )}\right )\right )+2 i a \left (1+e^{2 c}\right ) \left (d^2 x^4 \log \left (1-i e^{c+d x^2}\right )-d^2 x^4 \log \left (1+i e^{c+d x^2}\right )-2 d x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )+2 d x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )+2 \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )-2 \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )\right )\right )}{d^3 \left (1+e^{2 c}\right )}+\frac {3 b^2 x^4 \text {sech}(c) \sinh \left (d x^2\right )}{d}\right )}{6 \left (b+a \cosh \left (c+d x^2\right )\right )^2} \]
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\[\int x^{5} {\left (a +b \,\operatorname {sech}\left (d \,x^{2}+c \right )\right )}^{2}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1198 vs. \(2 (185) = 370\).
Time = 0.30 (sec) , antiderivative size = 1198, normalized size of antiderivative = 5.52 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^{5} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}\, dx \]
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\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{5} \,d x } \]
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\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^5\,{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2 \,d x \]
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