Optimal. Leaf size=261 \[ \frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a d^2 \sqrt{b^2-a^2}}-\frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a d^2 \sqrt{b^2-a^2}}+\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a d \sqrt{b^2-a^2}}+\frac{x^4}{4 a} \]
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Rubi [A] time = 0.539928, antiderivative size = 261, 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 = {4204, 4191, 3321, 2264, 2190, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a d^2 \sqrt{b^2-a^2}}-\frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a d^2 \sqrt{b^2-a^2}}+\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a d \sqrt{b^2-a^2}}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 4204
Rule 4191
Rule 3321
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3}{a+b \sec \left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b \sec (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 \cos (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac{x^4}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{x}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac{x^4}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a}\\ &=\frac{x^4}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt{-a^2+b^2}}+\frac{b \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt{-a^2+b^2}}\\ &=\frac{x^4}{4 a}+\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}-\frac{(i b) \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{i (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{(i b) \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{i (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{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{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^{i \left (c+d x^2\right )}\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^{i \left (c+d x^2\right )}\right )}{2 a \sqrt{-a^2+b^2} d^2}\\ &=\frac{x^4}{4 a}+\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d}+\frac{b \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{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^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a \sqrt{-a^2+b^2} d^2}\\ \end{align*}
Mathematica [B] time = 1.33764, size = 845, normalized size = 3.24 \[ \frac{\left (b+a \cos \left (d x^2+c\right )\right ) \left (x^4-\frac{2 b \left (2 \left (d x^2+c\right ) \tanh ^{-1}\left (\frac{(a+b) \cot \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )-2 \left (c+\cos ^{-1}\left (-\frac{b}{a}\right )\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \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 i \tanh ^{-1}\left (\frac{(a+b) \cot \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )+2 i \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{\sqrt{a^2-b^2} e^{-\frac{1}{2} i \left (d x^2+c\right )}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cos \left (d x^2+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 i \left (\tanh ^{-1}\left (\frac{(a+b) \cot \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )-\tanh ^{-1}\left (\frac{(a-b) \tan \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} i \left (d x^2+c\right )}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cos \left (d x^2+c\right )}}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )-2 i \tanh ^{-1}\left (\frac{(a-b) \tan \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 (i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right )}{a \left (a+b+\sqrt{a^2-b^2} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 i \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{(a+b) \left (-i a+i b+\sqrt{a^2-b^2}\right ) \left (\tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+i\right )}{a \left (a+b+\sqrt{a^2-b^2} \tan \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-\sqrt{a^2-b^2} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{a \left (a+b+\sqrt{a^2-b^2} \tan \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-\sqrt{a^2-b^2} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{a \left (a+b+\sqrt{a^2-b^2} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}\right )\right )\right )}{\sqrt{a^2-b^2} d^2}\right ) \sec \left (d x^2+c\right )}{4 a \left (a+b \sec \left (d x^2+c\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.11, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{a+b\sec \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 = 2.51719, size = 2566, normalized size = 9.83 \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 \frac{x^{3}}{a + b \sec{\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 \sec \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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