Optimal. Leaf size=120 \[ \frac {2}{3} a x^{3/2}-\frac {4 b x \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, 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, 5545,
4267, 2611, 2320, 6724} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2320
Rule 2611
Rule 4267
Rule 5545
Rule 6724
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) \text {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) \text {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(4 b) \text {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) \text {Subst}\left (\int \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(4 b) \text {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) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(4 b) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 7.29, size = 162, normalized size = 1.35 \begin {gather*} \frac {2 \left (a d^3 x^{3/2}-b 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 (1-e^{c+d \sqrt {x}}\right )+6 b d \sqrt {x} \text {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )+6 b d \sqrt {x} \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )+6 b \text {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )-6 b \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )\right )}{3 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.95, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 129, normalized size = 1.08 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {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.
[In]
[Out]
________________________________________________________________________________________