Optimal. Leaf size=120 \[ \frac {2}{3} a x^{3/2}+\frac {4 b \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \sqrt {x} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}-\frac {4 b x \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 120, 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, 5437, 4182, 2531, 2282, 6589} \[ -\frac {4 b \sqrt {x} \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2282
Rule 2531
Rule 4182
Rule 5437
Rule 6589
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx &=\int \left (a \sqrt {x}+b \sqrt {x} \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx\\ &=\frac {2}{3} a x^{3/2}+b \int \sqrt {x} \text {csch}\left (c+d \sqrt {x}\right ) \, dx\\ &=\frac {2}{3} a x^{3/2}+(2 b) \operatorname {Subst}\left (\int x^2 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {(4 b) \operatorname {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(4 b) \operatorname {Subst}\left (\int x \log \left (1+e^{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^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 b) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(4 b) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{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^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}\\ &=\frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 9.69, size = 142, normalized size = 1.18 \[ \frac {2 \left (a d^3 x^{3/2}+3 b d^2 x \log \left (1-e^{c+d \sqrt {x}}\right )-3 b d^2 x \log \left (e^{c+d \sqrt {x}}+1\right )-6 b d \sqrt {x} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )+6 b d \sqrt {x} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )+6 b \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )-6 b \text {Li}_3\left (e^{c+d \sqrt {x}}\right )\right )}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \sqrt {x} \operatorname {csch}\left (d \sqrt {x} + c\right ) + a \sqrt {x}, x\right ) \]
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 \[ \int {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right ) \sqrt {x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.64, size = 129, normalized size = 1.08 \[ \frac {2}{3} \, a x^{\frac {3}{2}} - \frac {2 \, {\left (\log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} + 2 \, {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{3}} + \frac {2 \, {\left (\log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} + 2 \, {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{3}} \]
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 \[ \int \sqrt {x}\,\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
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 \[ \int \sqrt {x} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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