Optimal. Leaf size=333 \[ \frac{12 i a b x \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{24 i a b \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{24 i a b \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{6 i b^2 \sqrt{x} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{3 b^2 \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}-\frac{2 i b^2 x^{3/2}}{d} \]
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Rubi [A] time = 0.431866, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4205, 4190, 4183, 2531, 6609, 2282, 6589, 4184, 3717, 2190} \[ \frac{12 i a b x \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{24 i a b \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{24 i a b \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{6 i b^2 \sqrt{x} \text{PolyLog}\left (2,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{3 b^2 \text{PolyLog}\left (3,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}-\frac{2 i b^2 x^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 4205
Rule 4190
Rule 4183
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4184
Rule 3717
Rule 2190
Rubi steps
\begin{align*} \int x \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b \csc (c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^3+2 a b x^3 \csc (c+d x)+b^2 x^3 \csc ^2(c+d x)\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^2}{2}+(4 a b) \operatorname{Subst}\left (\int x^3 \csc (c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int x^3 \csc ^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}-\frac{(12 a b) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(12 a b) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int x^2 \cot (c+d x) \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 i b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{12 i a b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{(24 i a b) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(24 i a b) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (12 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i (c+d x)} x^2}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 i b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{12 i a b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{(24 a b) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{(24 a b) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{2 i b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{12 i a b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b^2 \sqrt{x} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{(24 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(24 i a b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{2 i b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{12 i a b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b^2 \sqrt{x} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{24 i a b \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{24 i a b \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ &=-\frac{2 i b^2 x^{3/2}}{d}+\frac{a^2 x^2}{2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{2 b^2 x^{3/2} \cot \left (c+d \sqrt{x}\right )}{d}+\frac{6 b^2 x \log \left (1-e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^2}+\frac{12 i a b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 i a b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b^2 \sqrt{x} \text{Li}_2\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{24 a b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{3 b^2 \text{Li}_3\left (e^{2 i \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{24 i a b \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{24 i a b \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 8.22096, size = 449, normalized size = 1.35 \[ -\frac{2 i b \left (i \left (-6 i \left (a d^2 x+b d \sqrt{x}\right ) \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )+6 \left (b-2 a d \sqrt{x}\right ) \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )+12 a d \sqrt{x} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )-12 i a \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )+12 i a \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )+6 b \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )+2 a d^3 x^{3/2} \log \left (1-e^{i \left (c+d \sqrt{x}\right )}\right )-2 a d^3 x^{3/2} \log \left (1+e^{i \left (c+d \sqrt{x}\right )}\right )+3 b d^2 x \log \left (1-e^{i \left (c+d \sqrt{x}\right )}\right )+3 b d^2 x \log \left (1+e^{i \left (c+d \sqrt{x}\right )}\right )\right )+\left (6 b d \sqrt{x}-6 a d^2 x\right ) \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )+\frac{2 b e^{2 i c} d^3 x^{3/2}}{-1+e^{2 i c}}\right )}{d^4}+\frac{a^2 x^2}{2}+\frac{b^2 x^{3/2} \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d \sqrt{x}}{2}\right ) \csc \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{d}+\frac{b^2 x^{3/2} \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d \sqrt{x}}{2}\right ) \sec \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.182, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.65839, size = 2624, normalized size = 7.88 \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^{2} x \csc \left (d \sqrt{x} + c\right )^{2} + 2 \, a b x \csc \left (d \sqrt{x} + c\right ) + a^{2} 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 \csc{\left (c + d \sqrt{x} \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 \csc \left (d \sqrt{x} + c\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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