Integrand size = 22, antiderivative size = 959 \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {4 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {4 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )} \]
[Out]
Time = 1.29 (sec) , antiderivative size = 959, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5545, 4276, 3405, 3403, 2296, 2221, 2611, 2320, 6724, 5680, 2317, 2438} \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {4 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {4 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 x b^2}{a^2 \left (a^2+b^2\right ) d}+\frac {4 \sqrt {x} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {4 \sqrt {x} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {4 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 x \cosh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {4 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}+\frac {4 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}-\frac {8 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {8 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {8 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {8 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}+\frac {2 x^{3/2}}{3 a^2} \]
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3403
Rule 3405
Rule 4276
Rule 5545
Rule 5680
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2}{(a+b \text {csch}(c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^2}{a^2}+\frac {b^2 x^2}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{3/2}}{3 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^2}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{(b+a \sinh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = \frac {2 x^{3/2}}{3 a^2}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {x^2}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {x \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right ) d} \\ & = -\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2}}+\frac {(8 b) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2}}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right ) d}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right ) d} \\ & = -\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right )^{3/2}}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right )^{3/2}}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {(8 b) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {(8 b) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d} \\ & = -\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {(8 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {(8 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d} \\ & = -\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {(8 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {(8 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2} \\ & = -\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3} \\ & = -\frac {2 b^2 x}{a^2 \left (a^2+b^2\right ) d}+\frac {2 x^{3/2}}{3 a^2}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \sqrt {x} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {4 b^3 \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {8 b \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {4 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {4 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {8 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^2 x \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )} \\ \end{align*}
Time = 3.17 (sec) , antiderivative size = 948, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {\text {csch}^2\left (c+d \sqrt {x}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right ) \left (2 x^{3/2} \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )-\frac {6 b e^c \left (2 b e^c x-\frac {e^{-c} \left (-1+e^{2 c}\right ) \left (2 b d \sqrt {\left (a^2+b^2\right ) e^{2 c}} \sqrt {x} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 a^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-b^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 b d \sqrt {\left (a^2+b^2\right ) e^{2 c}} \sqrt {x} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 a^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+b^2 d^2 e^c x \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \left (b \sqrt {\left (a^2+b^2\right ) e^{2 c}}-2 a^2 d e^c \sqrt {x}-b^2 d e^c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \left (b \sqrt {\left (a^2+b^2\right ) e^{2 c}}+2 a^2 d e^c \sqrt {x}+b^2 d e^c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+4 a^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 b^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-4 a^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 b^2 e^c \operatorname {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{d^2 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}{\left (a^2+b^2\right ) d \left (-1+e^{2 c}\right )}+\frac {6 b^2 x \text {csch}(c) \left (b \cosh (c)+a \sinh \left (d \sqrt {x}\right )\right )}{\left (a^2+b^2\right ) d}\right )}{3 a^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \]
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\[\int \frac {\sqrt {x}}{\left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]
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\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {\sqrt {x}}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {\sqrt {x}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]
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