Optimal. Leaf size=526 \[ \frac{6 i b d^2 (c+d x) \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f^3 \sqrt{b^2-a^2}}-\frac{6 i b d^2 (c+d x) \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f^3 \sqrt{b^2-a^2}}+\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f^2 \sqrt{b^2-a^2}}-\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f^2 \sqrt{b^2-a^2}}-\frac{6 b d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f^4 \sqrt{b^2-a^2}}+\frac{6 b d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f^4 \sqrt{b^2-a^2}}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f \sqrt{b^2-a^2}}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f \sqrt{b^2-a^2}}+\frac{(c+d x)^4}{4 a d} \]
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Rubi [A] time = 1.04345, antiderivative size = 526, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4191, 3321, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{6 i b d^2 (c+d x) \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f^3 \sqrt{b^2-a^2}}-\frac{6 i b d^2 (c+d x) \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f^3 \sqrt{b^2-a^2}}+\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f^2 \sqrt{b^2-a^2}}-\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f^2 \sqrt{b^2-a^2}}-\frac{6 b d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f^4 \sqrt{b^2-a^2}}+\frac{6 b d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f^4 \sqrt{b^2-a^2}}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a f \sqrt{b^2-a^2}}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a f \sqrt{b^2-a^2}}+\frac{(c+d x)^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 4191
Rule 3321
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{a+b \sec (e+f x)} \, dx &=\int \left (\frac{(c+d x)^3}{a}-\frac{b (c+d x)^3}{a (b+a \cos (e+f x))}\right ) \, dx\\ &=\frac{(c+d x)^4}{4 a d}-\frac{b \int \frac{(c+d x)^3}{b+a \cos (e+f x)} \, dx}{a}\\ &=\frac{(c+d x)^4}{4 a d}-\frac{(2 b) \int \frac{e^{i (e+f x)} (c+d x)^3}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a}\\ &=\frac{(c+d x)^4}{4 a d}-\frac{(2 b) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{\sqrt{-a^2+b^2}}+\frac{(2 b) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{\sqrt{-a^2+b^2}}\\ &=\frac{(c+d x)^4}{4 a d}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}-\frac{(3 i b d) \int (c+d x)^2 \log \left (1+\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a \sqrt{-a^2+b^2} f}+\frac{(3 i b d) \int (c+d x)^2 \log \left (1+\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a \sqrt{-a^2+b^2} f}\\ &=\frac{(c+d x)^4}{4 a d}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}-\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}-\frac{\left (6 b d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a \sqrt{-a^2+b^2} f^2}+\frac{\left (6 b d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a \sqrt{-a^2+b^2} f^2}\\ &=\frac{(c+d x)^4}{4 a d}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}-\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}+\frac{6 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^3}-\frac{6 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^3}-\frac{\left (6 i b d^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a \sqrt{-a^2+b^2} f^3}+\frac{\left (6 i b d^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a \sqrt{-a^2+b^2} f^3}\\ &=\frac{(c+d x)^4}{4 a d}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}-\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}+\frac{6 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^3}-\frac{6 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^3}-\frac{\left (6 b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a \sqrt{-a^2+b^2} f^4}+\frac{\left (6 b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a \sqrt{-a^2+b^2} f^4}\\ &=\frac{(c+d x)^4}{4 a d}+\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}-\frac{i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}-\frac{3 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^2}+\frac{6 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^3}-\frac{6 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^3}-\frac{6 b d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^4}+\frac{6 b d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} f^4}\\ \end{align*}
Mathematica [A] time = 1.10002, size = 449, normalized size = 0.85 \[ \frac{\sec (e+f x) (a \cos (e+f x)+b) \left (x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+\frac{4 i b \left (\frac{3 i d \left (f^2 (c+d x)^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )+2 i d f (c+d x) \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )-2 d^2 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )\right )}{f^3}+\frac{3 d \left (2 d \left (f (c+d x) \text{PolyLog}\left (3,\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right )+i d \text{PolyLog}\left (4,\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right )\right )-i f^2 (c+d x)^2 \text{PolyLog}\left (2,\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right )\right )}{f^3}+(c+d x)^3 \log \left (1-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right )-(c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )\right )}{f \sqrt{b^2-a^2}}\right )}{4 a (a+b \sec (e+f x))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.566, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{3}}{a+b\sec \left ( fx+e \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 [C] time = 3.40952, size = 5515, normalized size = 10.48 \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{\left (c + d x\right )^{3}}{a + b \sec{\left (e + f x \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{{\left (d x + c\right )}^{3}}{b \sec \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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