Optimal. Leaf size=242 \[ \frac{2 i a b x^2 \text{PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 i a b x^2 \text{PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 a b \text{PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac{2 a b \text{PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )}{d^3}-\frac{i b^2 \text{PolyLog}\left (2,-e^{2 i \left (c+d x^2\right )}\right )}{2 d^3}+\frac{a^2 x^6}{6}-\frac{2 i a b x^4 \tan ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac{b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac{b^2 x^4 \tan \left (c+d x^2\right )}{2 d}-\frac{i b^2 x^4}{2 d} \]
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Rubi [A] time = 0.370284, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {4204, 4190, 4181, 2531, 2282, 6589, 4184, 3719, 2190, 2279, 2391} \[ \frac{2 i a b x^2 \text{PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 i a b x^2 \text{PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 a b \text{PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac{2 a b \text{PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )}{d^3}-\frac{i b^2 \text{PolyLog}\left (2,-e^{2 i \left (c+d x^2\right )}\right )}{2 d^3}+\frac{a^2 x^6}{6}-\frac{2 i a b x^4 \tan ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac{b^2 x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac{b^2 x^4 \tan \left (c+d x^2\right )}{2 d}-\frac{i b^2 x^4}{2 d} \]
Antiderivative was successfully verified.
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Rule 4204
Rule 4190
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4184
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^5 \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b \sec (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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) \operatorname{Subst}\left (\int x^2 \sec (c+d x) \, dx,x,x^2\right )+\frac{1}{2} b^2 \operatorname{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 \tan ^{-1}\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) \operatorname{Subst}\left (\int x \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}+\frac{(2 a b) \operatorname{Subst}\left (\int x \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}-\frac{b^2 \operatorname{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 \tan ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac{2 i a b x^2 \text{Li}_2\left (-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 i a b x^2 \text{Li}_2\left (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) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}+\frac{(2 i a b) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}+\frac{\left (2 i b^2\right ) \operatorname{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 \tan ^{-1}\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 \text{Li}_2\left (-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 i a b x^2 \text{Li}_2\left (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) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^3}-\frac{b^2 \operatorname{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 \tan ^{-1}\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 \text{Li}_2\left (-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 i a b x^2 \text{Li}_2\left (i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 a b \text{Li}_3\left (-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac{2 a b \text{Li}_3\left (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 ) \operatorname{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 \tan ^{-1}\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 \text{Li}_2\left (-i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{2 i a b x^2 \text{Li}_2\left (i e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac{i b^2 \text{Li}_2\left (-e^{2 i \left (c+d x^2\right )}\right )}{2 d^3}-\frac{2 a b \text{Li}_3\left (-i e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac{2 a b \text{Li}_3\left (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*}
Mathematica [A] time = 0.589636, size = 229, normalized size = 0.95 \[ \frac{12 i a b d x^2 \text{PolyLog}\left (2,-i e^{i \left (c+d x^2\right )}\right )-12 i a b d x^2 \text{PolyLog}\left (2,i e^{i \left (c+d x^2\right )}\right )-12 a b \text{PolyLog}\left (3,-i e^{i \left (c+d x^2\right )}\right )+12 a b \text{PolyLog}\left (3,i e^{i \left (c+d x^2\right )}\right )-3 i b^2 \text{PolyLog}\left (2,-e^{2 i \left (c+d x^2\right )}\right )+a^2 d^3 x^6-12 i a b d^2 x^4 \tan ^{-1}\left (e^{i \left (c+d x^2\right )}\right )+3 b^2 d^2 x^4 \tan \left (c+d x^2\right )+6 b^2 d x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )-3 i b^2 d^2 x^4}{6 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.206, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( a+b\sec \left ( d{x}^{2}+c \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a^{2} x^{6} + \frac{b^{2} x^{4} \sin \left (2 \, d x^{2} + 2 \, c\right ) + 4 \,{\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )} \int \frac{a b d x^{5} \cos \left (2 \, d x^{2} + 2 \, c\right ) \cos \left (d x^{2} + c\right ) + a b d x^{5} \cos \left (d x^{2} + c\right ) +{\left (a b d x^{5} \sin \left (d x^{2} + c\right ) - b^{2} x^{3}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d}\,{d x}}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.29686, size = 1960, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \left (a + b \sec{\left (c + d x^{2} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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