Integrand size = 20, antiderivative size = 120 \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2}{3} a x^{3/2}-\frac {4 b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {14, 5545, 4267, 2611, 2320, 6724} \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2}{3} a x^{3/2}-\frac {4 b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}+\frac {4 b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2} \]
[In]
[Out]
Rule 14
Rule 2320
Rule 2611
Rule 4267
Rule 5545
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \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 \text {arctanh}\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 \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(4 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = \frac {2}{3} a x^{3/2}-\frac {4 b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(4 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3} \\ & = \frac {2}{3} a x^{3/2}-\frac {4 b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.18 \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2 \left (a d^3 x^{3/2}+3 b d^2 x \log \left (1-e^{c+d \sqrt {x}}\right )-3 b d^2 x \log \left (1+e^{c+d \sqrt {x}}\right )-6 b d \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )+6 b d \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )+6 b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )-6 b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )\right )}{3 d^3} \]
[In]
[Out]
\[\int \left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right ) \sqrt {x}d x\]
[In]
[Out]
\[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
[In]
[Out]
\[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int \sqrt {x} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )\, dx \]
[In]
[Out]
none
Time = 0.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2}{3} \, a x^{\frac {3}{2}} - \frac {2 \, {\left (\log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} + 2 \, {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{3}} + \frac {2 \, {\left (\log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} + 2 \, {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{3}} \]
[In]
[Out]
\[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int \sqrt {x}\,\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
[In]
[Out]