Integrand size = 20, antiderivative size = 1983 \[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=-\frac {2 b^2 x^{5/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^3}{3 a^2}+\frac {10 b^2 x^2 \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^{5/2} \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^{5/2} \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 {10 b^2 x^2 \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^{5/2} \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^{5/2} \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 {40 b^2 x^{3/2} \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 {10 b^3 x^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 )^{3/2} d^2}-\frac {20 b x^2 \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 {40 b^2 x^{3/2} \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 {10 b^3 x^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 )^{3/2} d^2}+\frac {20 b x^2 \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 {120 b^2 x \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 ) d^4}-\frac {40 b^3 x^{3/2} \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 {80 b x^{3/2} \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 {120 b^2 x \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 ) d^4}+\frac {40 b^3 x^{3/2} \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 {80 b x^{3/2} \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 {240 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^5}+\frac {120 b^3 x \operatorname {PolyLog}\left (4,-\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^4}-\frac {240 b x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}+\frac {240 b^2 \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^5}-\frac {120 b^3 x \operatorname {PolyLog}\left (4,-\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^4}+\frac {240 b x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {240 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^6}-\frac {240 b^3 \sqrt {x} \operatorname {PolyLog}\left (5,-\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^5}+\frac {480 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^5}-\frac {240 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^6}+\frac {240 b^3 \sqrt {x} \operatorname {PolyLog}\left (5,-\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^5}-\frac {480 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^5}+\frac {240 b^3 \operatorname {PolyLog}\left (6,-\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^6}-\frac {480 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^6}-\frac {240 b^3 \operatorname {PolyLog}\left (6,-\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^6}+\frac {480 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^6}-\frac {2 b^2 x^{5/2} \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 = 2.01 (sec) , antiderivative size = 1983, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5545, 4276, 3405, 3403, 2296, 2221, 2611, 6744, 2320, 6724, 5680} \[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 x^{5/2} \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^{5/2} \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 {10 x^2 \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 {10 x^2 \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 {40 x^{3/2} \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 {40 x^{3/2} \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 {120 x \operatorname {PolyLog}\left (4,-\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^4}-\frac {120 x \operatorname {PolyLog}\left (4,-\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^4}-\frac {240 \sqrt {x} \operatorname {PolyLog}\left (5,-\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^5}+\frac {240 \sqrt {x} \operatorname {PolyLog}\left (5,-\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^5}+\frac {240 \operatorname {PolyLog}\left (6,-\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^6}-\frac {240 \operatorname {PolyLog}\left (6,-\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^6}-\frac {2 x^{5/2} b^2}{a^2 \left (a^2+b^2\right ) d}+\frac {10 x^2 \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 {10 x^2 \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 {40 x^{3/2} \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 {40 x^{3/2} \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 {120 x \operatorname {PolyLog}\left (3,-\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^4}-\frac {120 x \operatorname {PolyLog}\left (3,-\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^4}+\frac {240 \sqrt {x} \operatorname {PolyLog}\left (4,-\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^5}+\frac {240 \sqrt {x} \operatorname {PolyLog}\left (4,-\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^5}-\frac {240 \operatorname {PolyLog}\left (5,-\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^6}-\frac {240 \operatorname {PolyLog}\left (5,-\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^6}-\frac {2 x^{5/2} \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^{5/2} \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^{5/2} \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 {20 x^2 \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 {20 x^2 \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 {80 x^{3/2} \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 {80 x^{3/2} \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 {240 x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {240 x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {480 \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^5}-\frac {480 \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^5}-\frac {480 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^6}+\frac {480 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^6}+\frac {x^3}{3 a^2} \]
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3403
Rule 3405
Rule 4276
Rule 5545
Rule 5680
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^5}{(a+b \text {csch}(c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^5}{a^2}+\frac {b^2 x^5}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x^5}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^3}{3 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^5}{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^5}{(b+a \sinh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = \frac {x^3}{3 a^2}-\frac {2 b^2 x^{5/2} \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^5}{-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^5}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )}+\frac {\left (10 b^2\right ) \text {Subst}\left (\int \frac {x^4 \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^{5/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^3}{3 a^2}-\frac {2 b^2 x^{5/2} \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^5}{-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^5}{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^5}{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 (10 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{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 (10 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{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^{5/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^3}{3 a^2}+\frac {10 b^2 x^2 \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^{5/2} \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 {10 b^2 x^2 \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^{5/2} \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^{5/2} \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^5}{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^5}{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 (40 b^2\right ) \text {Subst}\left (\int x^3 \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 (40 b^2\right ) \text {Subst}\left (\int x^3 \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 {(20 b) \text {Subst}\left (\int x^4 \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 {(20 b) \text {Subst}\left (\int x^4 \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^{5/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^3}{3 a^2}+\frac {10 b^2 x^2 \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^{5/2} \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^{5/2} \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 {10 b^2 x^2 \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^{5/2} \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^{5/2} \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 {40 b^2 x^{3/2} \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 {20 b x^2 \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 {40 b^2 x^{3/2} \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 {20 b x^2 \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^{5/2} \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 (120 b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-\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^3}-\frac {\left (120 b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-\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^3}+\frac {(80 b) \text {Subst}\left (\int x^3 \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 {(80 b) \text {Subst}\left (\int x^3 \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 (10 b^3\right ) \text {Subst}\left (\int x^4 \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 (10 b^3\right ) \text {Subst}\left (\int x^4 \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} \\ & = \text {Too large to display} \\ \end{align*}
Time = 6.12 (sec) , antiderivative size = 2085, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {x^{2}}{\left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]
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\[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^2}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]
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