Optimal. Leaf size=144 \[ \frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {4 i b \sqrt {x} \text {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \text {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 b \text {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b \text {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {14, 4290,
4268, 2611, 2320, 6724} \begin {gather*} \frac {2}{3} a x^{3/2}-\frac {4 b \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 i b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2320
Rule 2611
Rule 4268
Rule 4290
Rule 6724
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx &=\int \left (a \sqrt {x}+b \sqrt {x} \csc \left (c+d \sqrt {x}\right )\right ) \, dx\\ &=\frac {2}{3} a x^{3/2}+b \int \sqrt {x} \csc \left (c+d \sqrt {x}\right ) \, dx\\ &=\frac {2}{3} a x^{3/2}+(2 b) \text {Subst}\left (\int x^2 \csc (c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {(4 b) \text {Subst}\left (\int x \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(4 b) \text {Subst}\left (\int x \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {4 i b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(4 i b) \text {Subst}\left (\int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(4 i b) \text {Subst}\left (\int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {4 i b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(4 b) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(4 b) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}\\ &=\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {4 i b \sqrt {x} \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 b \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 4.25, size = 191, normalized size = 1.33 \begin {gather*} \frac {2 \left (a d^3 x^{3/2}-6 b d^2 x \tanh ^{-1}\left (\cos \left (c+d \sqrt {x}\right )+i \sin \left (c+d \sqrt {x}\right )\right )+6 i b d \sqrt {x} \text {PolyLog}\left (2,-\cos \left (c+d \sqrt {x}\right )-i \sin \left (c+d \sqrt {x}\right )\right )-6 i b d \sqrt {x} \text {PolyLog}\left (2,\cos \left (c+d \sqrt {x}\right )+i \sin \left (c+d \sqrt {x}\right )\right )-6 b \text {PolyLog}\left (3,-\cos \left (c+d \sqrt {x}\right )-i \sin \left (c+d \sqrt {x}\right )\right )+6 b \text {PolyLog}\left (3,\cos \left (c+d \sqrt {x}\right )+i \sin \left (c+d \sqrt {x}\right )\right )\right )}{3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (a +b \csc \left (c +d \sqrt {x}\right )\right ) \sqrt {x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 370 vs. \(2 (110) = 220\).
time = 0.32, size = 370, normalized size = 2.57 \begin {gather*} \frac {2 \, {\left (d \sqrt {x} + c\right )}^{3} a - 6 \, {\left (d \sqrt {x} + c\right )}^{2} a c + 6 \, {\left (d \sqrt {x} + c\right )} a c^{2} - 6 \, b c^{2} \log \left (\cot \left (d \sqrt {x} + c\right ) + \csc \left (d \sqrt {x} + c\right )\right ) + 6 \, {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} b + 2 i \, {\left (d \sqrt {x} + c\right )} b c\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), \cos \left (d \sqrt {x} + c\right ) + 1\right ) + 6 \, {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} b + 2 i \, {\left (d \sqrt {x} + c\right )} b c\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), -\cos \left (d \sqrt {x} + c\right ) + 1\right ) + 12 \, {\left (i \, {\left (d \sqrt {x} + c\right )} b - i \, b c\right )} {\rm Li}_2\left (-e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) + 12 \, {\left (-i \, {\left (d \sqrt {x} + c\right )} b + i \, b c\right )} {\rm Li}_2\left (e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) - 3 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} b - 2 \, {\left (d \sqrt {x} + c\right )} b c\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} + 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right ) + 3 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} b - 2 \, {\left (d \sqrt {x} + c\right )} b c\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} - 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right ) - 12 \, b {\rm Li}_{3}(-e^{\left (i \, d \sqrt {x} + i \, c\right )}) + 12 \, b {\rm Li}_{3}(e^{\left (i \, d \sqrt {x} + i \, c\right )})}{3 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {x}\,\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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