Optimal. Leaf size=164 \[ \frac {a x^2}{2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b x \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 b \sqrt {x} \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \text {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \text {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4} \]
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Rubi [A]
time = 0.12, antiderivative size = 164, normalized size of antiderivative = 1.00, 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, 5545,
4267, 2611, 6744, 2320, 6724} \begin {gather*} \frac {a x^2}{2}-\frac {12 b \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \sqrt {x} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {6 b x \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2320
Rule 2611
Rule 4267
Rule 5545
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx &=\int \left (a x+b x \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx\\ &=\frac {a x^2}{2}+b \int x \text {csch}\left (c+d \sqrt {x}\right ) \, dx\\ &=\frac {a x^2}{2}+(2 b) \text {Subst}\left (\int x^3 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {a x^2}{2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {(6 b) \text {Subst}\left (\int x^2 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(6 b) \text {Subst}\left (\int x^2 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {a x^2}{2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b x \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(12 b) \text {Subst}\left (\int x \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(12 b) \text {Subst}\left (\int x \text {Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {a x^2}{2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b x \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 b \sqrt {x} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(12 b) \text {Subst}\left (\int \text {Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(12 b) \text {Subst}\left (\int \text {Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}\\ &=\frac {a x^2}{2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b x \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 b \sqrt {x} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(12 b) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(12 b) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^4}\\ &=\frac {a x^2}{2}-\frac {4 b x^{3/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b x \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 b \sqrt {x} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}\\ \end {align*}
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Mathematica [A]
time = 1.89, size = 181, normalized size = 1.10 \begin {gather*} \frac {a x^2}{2}+\frac {2 b \left (d^3 x^{3/2} \log \left (1-e^{c+d \sqrt {x}}\right )-d^3 x^{3/2} \log \left (1+e^{c+d \sqrt {x}}\right )-3 d^2 x \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )+3 d^2 x \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )+6 d \sqrt {x} \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )-6 d \sqrt {x} \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )-6 \text {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )+6 \text {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )\right )}{d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.25, size = 0, normalized size = 0.00 \[\int x \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 173, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, a x^{2} - \frac {2 \, {\left (\log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{3} + 3 \, {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} - 6 \, \log \left (e^{\left (d \sqrt {x}\right )}\right ) {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{4}} + \frac {2 \, {\left (\log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{3} + 3 \, {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} - 6 \, \log \left (e^{\left (d \sqrt {x}\right )}\right ) {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{4}} \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 x \left (a + b \operatorname {csch}{\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 x\,\left (a+\frac {b}{\mathrm {sinh}\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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