Optimal. Leaf size=241 \[ -\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}\right )}{2 a d^2 \sqrt{b^2-a^2}}+\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )}{2 a d^2 \sqrt{b^2-a^2}}-\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )}{2 a d \sqrt{b^2-a^2}}+\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )}{2 a d \sqrt{b^2-a^2}}+\frac{x^4}{4 a} \]
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Rubi [A] time = 0.508309, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5436, 4191, 3320, 2264, 2190, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}\right )}{2 a d^2 \sqrt{b^2-a^2}}+\frac{b \text{PolyLog}\left (2,-\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}\right )}{2 a d^2 \sqrt{b^2-a^2}}-\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{b-\sqrt{b^2-a^2}}+1\right )}{2 a d \sqrt{b^2-a^2}}+\frac{b x^2 \log \left (\frac{a e^{c+d x^2}}{\sqrt{b^2-a^2}+b}+1\right )}{2 a d \sqrt{b^2-a^2}}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 5436
Rule 4191
Rule 3320
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3}{a+b \text{sech}\left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b \text{sech}(c+d x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{x}{a}-\frac{b x}{a (b+a \cosh (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac{x^4}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{x}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac{x^4}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a}\\ &=\frac{x^4}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt{-a^2+b^2}}+\frac{b \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt{-a^2+b^2}}\\ &=\frac{x^4}{4 a}-\frac{b x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}+\frac{b x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt{-a^2+b^2} d}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt{-a^2+b^2} d}\\ &=\frac{x^4}{4 a}-\frac{b x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}+\frac{b x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a \sqrt{-a^2+b^2} d^2}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a \sqrt{-a^2+b^2} d^2}\\ &=\frac{x^4}{4 a}-\frac{b x^2 \log \left (1+\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}+\frac{b x^2 \log \left (1+\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}-\frac{b \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d^2}+\frac{b \text{Li}_2\left (-\frac{a e^{c+d x^2}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d^2}\\ \end{align*}
Mathematica [C] time = 1.67337, size = 843, normalized size = 3.5 \[ \frac{\left (b+a \cosh \left (d x^2+c\right )\right ) \left (x^4+\frac{2 b \left (2 \left (d x^2+c\right ) \tan ^{-1}\left (\frac{(a+b) \coth \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )+2 \left (c-i \cos ^{-1}\left (-\frac{b}{a}\right )\right ) \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 \left (\tan ^{-1}\left (\frac{(a+b) \coth \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )+\tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right )\right ) \log \left (\frac{\sqrt{a^2-b^2} e^{-\frac{d x^2}{2}-\frac{c}{2}}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cosh \left (d x^2+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )-2 \left (\tan ^{-1}\left (\frac{(a+b) \coth \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )+\tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right )\right ) \log \left (\frac{\sqrt{a^2-b^2} e^{\frac{1}{2} \left (d x^2+c\right )}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cosh \left (d x^2+c\right )}}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{(a+b) \left (-a+b+i \sqrt{a^2-b^2}\right ) \left (\tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )-1\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )-2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{(a+b) \left (a-b+i \sqrt{a^2-b^2}\right ) \left (\tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (b-i \sqrt{a^2-b^2}\right ) \left (a+b-i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (b+i \sqrt{a^2-b^2}\right ) \left (a+b-i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}\right )\right )\right )}{\sqrt{a^2-b^2} d^2}\right ) \text{sech}\left (d x^2+c\right )}{4 a \left (a+b \text{sech}\left (d x^2+c\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{a+b{\rm sech} \left (d{x}^{2}+c\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24301, size = 1230, normalized size = 5.1 \begin{align*} \frac{{\left (a^{2} - b^{2}\right )} d^{2} x^{4} + 2 \, a b c \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) + 2 \, a \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, a b c \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) - 2 \, a \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, a b \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}}{\rm Li}_2\left (-\frac{b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) +{\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} + a}{a} + 1\right ) - 2 \, a b \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}}{\rm Li}_2\left (-\frac{b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) -{\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} + a}{a} + 1\right ) + 2 \,{\left (a b d x^{2} + a b c\right )} \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} \log \left (\frac{b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) +{\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} + a}{a}\right ) - 2 \,{\left (a b d x^{2} + a b c\right )} \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} \log \left (\frac{b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) -{\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt{-\frac{a^{2} - b^{2}}{a^{2}}} + a}{a}\right )}{4 \,{\left (a^{3} - a b^{2}\right )} d^{2}} \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 \frac{x^{3}}{a + b \operatorname{sech}{\left (c + d x^{2} \right )}}\, 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 \frac{x^{3}}{b \operatorname{sech}\left (d x^{2} + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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