Integrand size = 18, antiderivative size = 242 \[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=-\frac {i b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d x^2\right )}\right )}{2 d^3}-\frac {2 a b \operatorname {PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d} \]
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Time = 0.47 (sec) , antiderivative size = 242, 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 = {4289, 4275, 4266, 2611, 2320, 6724, 4269, 3800, 2221, 2317, 2438} \[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {2 a b \operatorname {PolyLog}\left (3,-i e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,i e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \left (d x^2+c\right )}\right )}{2 d^3}+\frac {b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d}-\frac {i b^2 x^4}{2 d} \]
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3800
Rule 4266
Rule 4269
Rule 4275
Rule 4289
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (a+b \sec (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 \sec (c+d x)+b^2 x^2 \sec ^2(c+d x)\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^6}{6}+(a b) \text {Subst}\left (\int x^2 \sec (c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \text {Subst}\left (\int x^2 \sec ^2(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d}-\frac {(2 a b) \text {Subst}\left (\int x \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}+\frac {(2 a b) \text {Subst}\left (\int x \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}-\frac {b^2 \text {Subst}\left (\int x \tan (c+d x) \, dx,x,x^2\right )}{d} \\ & = -\frac {i b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d}-\frac {(2 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i (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^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x}{1+e^{2 i (c+d x)}} \, dx,x,x^2\right )}{d} \\ & = -\frac {i b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d}-\frac {(2 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {(2 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^3}-\frac {b^2 \text {Subst}\left (\int \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,x^2\right )}{d^2} \\ & = -\frac {i b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b \operatorname {PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (c+d x^2\right )}\right )}{2 d^3} \\ & = -\frac {i b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 i a b x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d x^2\right )}\right )}{2 d^3}-\frac {2 a b \operatorname {PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b^2 x^4 \tan \left (c+d x^2\right )}{2 d} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95 \[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\frac {-3 i b^2 d^2 x^4+a^2 d^3 x^6-12 i a b d^2 x^4 \arctan \left (e^{i \left (c+d x^2\right )}\right )+6 b^2 d x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )+12 i a b d x^2 \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )-12 i a b d x^2 \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )-3 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d x^2\right )}\right )-12 a b \operatorname {PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )+12 a b \operatorname {PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )+3 b^2 d^2 x^4 \tan \left (c+d x^2\right )}{6 d^3} \]
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\[\int x^{5} {\left (a +b \sec \left (d \,x^{2}+c \right )\right )}^{2}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (199) = 398\).
Time = 0.32 (sec) , antiderivative size = 799, normalized size of antiderivative = 3.30 \[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\frac {a^{2} d^{3} x^{6} \cos \left (d x^{2} + c\right ) + 3 \, b^{2} d^{2} x^{4} \sin \left (d x^{2} + c\right ) - 6 \, a b \cos \left (d x^{2} + c\right ) {\rm polylog}\left (3, i \, \cos \left (d x^{2} + c\right ) + \sin \left (d x^{2} + c\right )\right ) + 6 \, a b \cos \left (d x^{2} + c\right ) {\rm polylog}\left (3, i \, \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right )\right ) - 6 \, a b \cos \left (d x^{2} + c\right ) {\rm polylog}\left (3, -i \, \cos \left (d x^{2} + c\right ) + \sin \left (d x^{2} + c\right )\right ) + 6 \, a b \cos \left (d x^{2} + c\right ) {\rm polylog}\left (3, -i \, \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right )\right ) - 3 \, {\left (2 i \, a b d x^{2} - i \, b^{2}\right )} \cos \left (d x^{2} + c\right ) {\rm Li}_2\left (i \, \cos \left (d x^{2} + c\right ) + \sin \left (d x^{2} + c\right )\right ) - 3 \, {\left (2 i \, a b d x^{2} + i \, b^{2}\right )} \cos \left (d x^{2} + c\right ) {\rm Li}_2\left (i \, \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right )\right ) - 3 \, {\left (-2 i \, a b d x^{2} + i \, b^{2}\right )} \cos \left (d x^{2} + c\right ) {\rm Li}_2\left (-i \, \cos \left (d x^{2} + c\right ) + \sin \left (d x^{2} + c\right )\right ) - 3 \, {\left (-2 i \, a b d x^{2} - i \, b^{2}\right )} \cos \left (d x^{2} + c\right ) {\rm Li}_2\left (-i \, \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right )\right ) + 3 \, {\left (a b c^{2} - b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + i\right ) - 3 \, {\left (a b c^{2} + b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + i\right ) + 3 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (i \, \cos \left (d x^{2} + c\right ) + \sin \left (d x^{2} + c\right ) + 1\right ) - 3 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2} - a b c^{2} - b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (i \, \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right ) + 1\right ) + 3 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (-i \, \cos \left (d x^{2} + c\right ) + \sin \left (d x^{2} + c\right ) + 1\right ) - 3 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2} - a b c^{2} - b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (-i \, \cos \left (d x^{2} + c\right ) - \sin \left (d x^{2} + c\right ) + 1\right ) + 3 \, {\left (a b c^{2} - b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + i\right ) - 3 \, {\left (a b c^{2} + b^{2} c\right )} \cos \left (d x^{2} + c\right ) \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + i\right )}{6 \, d^{3} \cos \left (d x^{2} + c\right )} \]
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\[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\int x^{5} \left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}\, dx \]
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\[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x^{5} \,d x } \]
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\[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx=\int x^5\,{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2 \,d x \]
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