Optimal. Leaf size=2123 \[ \text {result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.10, antiderivative size = 2123, normalized size of antiderivative = 1.00, number of steps
used = 49, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5544, 4276,
3405, 3401, 2296, 2221, 2611, 6744, 2320, 6724, 5681} \begin {gather*} \frac {2 x^{5/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 x^{5/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {10 x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {10 x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {40 x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {40 x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {120 x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {120 x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {240 \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {240 \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {240 \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {240 \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^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 {b^2-a^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 {b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {40 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {40 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {120 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {120 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}-\frac {240 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}-\frac {240 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}+\frac {240 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^6}+\frac {240 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^6}+\frac {2 x^{5/2} \sinh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {4 x^{5/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {4 x^{5/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {20 x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {20 x^2 \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {80 x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {80 x^{3/2} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {240 x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {240 x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {480 \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}-\frac {480 \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}-\frac {480 \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^6}+\frac {480 \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^6}+\frac {x^3}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3401
Rule 3405
Rule 4276
Rule 5544
Rule 5681
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^5}{(a+b \text {sech}(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 \cosh (c+d x))^2}-\frac {2 b x^5}{a^2 (b+a \cosh (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 \cosh (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 \cosh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=\frac {x^3}{3 a^2}+\frac {2 b^2 x^{5/2} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \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 \cosh (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 \sinh (c+d x)}{b+a \cosh (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} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \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} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (120 b^2\right ) \text {Subst}\left (\int x^2 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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}\\ &=\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 {80 b x^{3/2} \text {Li}_3\left (-\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 \text {Li}_3\left (-\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 {80 b x^{3/2} \text {Li}_3\left (-\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^{5/2} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (240 b^2\right ) \text {Subst}\left (\int x \text {Li}_3\left (-\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^4}-\frac {\left (240 b^2\right ) \text {Subst}\left (\int x \text {Li}_3\left (-\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^4}-\frac {(240 b) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\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^3}+\frac {(240 b) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\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^3}-\frac {\left (40 b^3\right ) \text {Subst}\left (\int x^3 \text {Li}_2\left (-\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 (40 b^3\right ) \text {Subst}\left (\int x^3 \text {Li}_2\left (-\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^{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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_4\left (-\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 {240 b x \text {Li}_4\left (-\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} \text {Li}_4\left (-\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 {240 b x \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {2 b^2 x^{5/2} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (240 b^2\right ) \text {Subst}\left (\int \text {Li}_4\left (-\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^5}+\frac {\left (240 b^2\right ) \text {Subst}\left (\int \text {Li}_4\left (-\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^5}+\frac {(480 b) \text {Subst}\left (\int x \text {Li}_4\left (-\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^4}-\frac {(480 b) \text {Subst}\left (\int x \text {Li}_4\left (-\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^4}+\frac {\left (120 b^3\right ) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\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^3}-\frac {\left (120 b^3\right ) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\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^3}\\ &=\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {480 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}-\frac {480 b \sqrt {x} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {2 b^2 x^{5/2} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (240 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_4\left (\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^6}+\frac {\left (240 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_4\left (-\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^6}-\frac {(480 b) \text {Subst}\left (\int \text {Li}_5\left (-\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^5}+\frac {(480 b) \text {Subst}\left (\int \text {Li}_5\left (-\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^5}-\frac {\left (240 b^3\right ) \text {Subst}\left (\int x \text {Li}_4\left (-\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^4}+\frac {\left (240 b^3\right ) \text {Subst}\left (\int x \text {Li}_4\left (-\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^4}\\ &=\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_5\left (-\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} \text {Li}_5\left (-\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} \text {Li}_5\left (-\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 \text {Li}_5\left (-\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} \text {Li}_5\left (-\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} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {2 b^2 x^{5/2} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {(480 b) \text {Subst}\left (\int \frac {\text {Li}_5\left (\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^6}+\frac {(480 b) \text {Subst}\left (\int \frac {\text {Li}_5\left (-\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^6}+\frac {\left (240 b^3\right ) \text {Subst}\left (\int \text {Li}_5\left (-\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^5}-\frac {\left (240 b^3\right ) \text {Subst}\left (\int \text {Li}_5\left (-\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^5}\\ &=\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_5\left (-\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} \text {Li}_5\left (-\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} \text {Li}_5\left (-\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 \text {Li}_5\left (-\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} \text {Li}_5\left (-\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} \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}-\frac {480 b \text {Li}_6\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^6}+\frac {480 b \text {Li}_6\left (-\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} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (240 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_5\left (\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^6}-\frac {\left (240 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_5\left (-\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^6}\\ &=\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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 \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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} \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_4\left (-\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 \text {Li}_5\left (-\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} \text {Li}_5\left (-\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} \text {Li}_5\left (-\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 \text {Li}_5\left (-\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} \text {Li}_5\left (-\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} \text {Li}_5\left (-\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 \text {Li}_6\left (-\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 \text {Li}_6\left (-\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 \text {Li}_6\left (-\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 \text {Li}_6\left (-\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} \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 11.33, size = 2173, normalized size = 1.02 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 3.50, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 \frac {x^{2}}{\left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, 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 = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________