Optimal. Leaf size=220 \[ \frac{6 i b x \text{PolyLog}\left (2,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{PolyLog}\left (2,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{PolyLog}\left (4,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{PolyLog}\left (4,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d} \]
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Rubi [A] time = 0.186068, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {14, 4204, 4181, 2531, 6609, 2282, 6589} \[ \frac{6 i b x \text{PolyLog}\left (2,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{PolyLog}\left (2,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{PolyLog}\left (4,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{PolyLog}\left (4,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4204
Rule 4181
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \left (a+b \sec \left (c+d \sqrt{x}\right )\right ) \, dx &=\int \left (a x+b x \sec \left (c+d \sqrt{x}\right )\right ) \, dx\\ &=\frac{a x^2}{2}+b \int x \sec \left (c+d \sqrt{x}\right ) \, dx\\ &=\frac{a x^2}{2}+(2 b) \operatorname{Subst}\left (\int x^3 \sec (c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{(6 b) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(6 b) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}\\ &=\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{(12 i b) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(12 i b) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{Li}_3\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{(12 b) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{(12 b) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{Li}_3\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{(12 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(12 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ &=\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{Li}_3\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{Li}_4\left (-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{Li}_4\left (i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.0597768, size = 223, normalized size = 1.01 \[ \frac{6 i b x \text{PolyLog}\left (2,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{PolyLog}\left (2,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{PolyLog}\left (4,-i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{PolyLog}\left (4,i e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 i b x^{3/2} \tan ^{-1}\left (e^{i c+i d \sqrt{x}}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.103, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\sec \left ( c+d\sqrt{x} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.19077, size = 729, normalized size = 3.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x \sec \left (d \sqrt{x} + c\right ) + a x, x\right ) \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 \left (a + b \sec{\left (c + d \sqrt{x} \right )}\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{\left (b \sec \left (d \sqrt{x} + c\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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