Integrand size = 20, antiderivative size = 2851 \[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}-\frac {14 b^2 x^3 \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^{7/2} \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^{7/2} \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 {14 b^2 x^3 \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^{7/2} \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^{7/2} \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 {84 b^2 x^{5/2} \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 {14 b^3 x^3 \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 {28 b x^3 \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 {84 b^2 x^{5/2} \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 {14 b^3 x^3 \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 {28 b x^3 \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 {420 b^2 x^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 ) d^4}-\frac {84 b^3 x^{5/2} \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 {168 b x^{5/2} \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 {420 b^2 x^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 ) d^4}+\frac {84 b^3 x^{5/2} \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 {168 b x^{5/2} \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 {1680 b^2 x^{3/2} \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 {420 b^3 x^2 \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 {840 b x^2 \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 {1680 b^2 x^{3/2} \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 {420 b^3 x^2 \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 {840 b x^2 \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 {5040 b^2 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 ) d^6}-\frac {1680 b^3 x^{3/2} \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 )^{3/2} d^5}+\frac {3360 b x^{3/2} \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 {5040 b^2 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 ) d^6}+\frac {1680 b^3 x^{3/2} \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 )^{3/2} d^5}-\frac {3360 b x^{3/2} \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 {10080 b^2 \sqrt {x} \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 ) d^7}+\frac {5040 b^3 x \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 {10080 b x \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 {10080 b^2 \sqrt {x} \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 ) d^7}-\frac {5040 b^3 x \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 {10080 b x \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 {10080 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac {10080 b^3 \sqrt {x} \operatorname {PolyLog}\left (7,-\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^7}+\frac {20160 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^7}+\frac {10080 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac {10080 b^3 \sqrt {x} \operatorname {PolyLog}\left (7,-\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^7}-\frac {20160 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^7}+\frac {10080 b^3 \operatorname {PolyLog}\left (8,-\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^8}-\frac {20160 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^8}-\frac {10080 b^3 \operatorname {PolyLog}\left (8,-\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^8}+\frac {20160 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^8}+\frac {2 b^2 x^{7/2} \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 )} \]
[Out]
Time = 2.78 (sec) , antiderivative size = 2851, normalized size of antiderivative = 1.00, number of steps used = 61, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5544, 4276, 3405, 3401, 2296, 2221, 2611, 6744, 2320, 6724, 5681} \[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\frac {x^4}{4 a^2}-\frac {4 b \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a^2 \sqrt {b^2-a^2} d}+\frac {2 b^3 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {4 b \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a^2 \sqrt {b^2-a^2} d}-\frac {2 b^3 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {2 b^2 \sinh \left (c+d \sqrt {x}\right ) x^{7/2}}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {2 b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}-\frac {14 b^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) x^3}{a^2 \left (a^2-b^2\right ) d^2}-\frac {14 b^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) x^3}{a^2 \left (a^2-b^2\right ) d^2}-\frac {28 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a^2 \sqrt {b^2-a^2} d^2}+\frac {14 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac {28 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a^2 \sqrt {b^2-a^2} d^2}-\frac {14 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {84 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \left (a^2-b^2\right ) d^3}-\frac {84 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \left (a^2-b^2\right ) d^3}+\frac {168 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \sqrt {b^2-a^2} d^3}-\frac {84 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {168 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \sqrt {b^2-a^2} d^3}+\frac {84 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {420 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {420 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (a^2-b^2\right ) d^4}-\frac {840 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^4}+\frac {420 b^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {840 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^4}-\frac {420 b^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {1680 b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \left (a^2-b^2\right ) d^5}-\frac {1680 b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \left (a^2-b^2\right ) d^5}+\frac {3360 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \sqrt {b^2-a^2} d^5}-\frac {1680 b^3 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^5}-\frac {3360 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \sqrt {b^2-a^2} d^5}+\frac {1680 b^3 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {5040 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \left (a^2-b^2\right ) d^6}+\frac {5040 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \left (a^2-b^2\right ) d^6}-\frac {10080 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \sqrt {b^2-a^2} d^6}+\frac {5040 b^3 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \left (b^2-a^2\right )^{3/2} d^6}+\frac {10080 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \sqrt {b^2-a^2} d^6}-\frac {5040 b^3 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {10080 b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (a^2-b^2\right ) d^7}-\frac {10080 b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (a^2-b^2\right ) d^7}+\frac {20160 b \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \sqrt {b^2-a^2} d^7}-\frac {10080 b^3 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (b^2-a^2\right )^{3/2} d^7}-\frac {20160 b \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \sqrt {b^2-a^2} d^7}+\frac {10080 b^3 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (b^2-a^2\right )^{3/2} d^7}+\frac {10080 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac {10080 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac {20160 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^8}+\frac {10080 b^3 \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^8}+\frac {20160 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^8}-\frac {10080 b^3 \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^8} \]
[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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{(a+b \text {sech}(c+d x))^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^7}{a^2}+\frac {b^2 x^7}{a^2 (b+a \cosh (c+d x))^2}-\frac {2 b x^7}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^4}{4 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^7}{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^7}{(b+a \cosh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = \frac {x^4}{4 a^2}+\frac {2 b^2 x^{7/2} \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^7}{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^7}{b+a \cosh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}-\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {x^6 \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^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}+\frac {2 b^2 x^{7/2} \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^7}{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^7}{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^7}{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 (14 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^6}{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 (14 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^6}{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^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}-\frac {14 b^2 x^3 \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^{7/2} \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 {14 b^2 x^3 \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^{7/2} \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^{7/2} \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^7}{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^7}{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 (84 b^2\right ) \text {Subst}\left (\int x^5 \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 (84 b^2\right ) \text {Subst}\left (\int x^5 \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 {(28 b) \text {Subst}\left (\int x^6 \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 {(28 b) \text {Subst}\left (\int x^6 \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^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^4}{4 a^2}-\frac {14 b^2 x^3 \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^{7/2} \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^{7/2} \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 {14 b^2 x^3 \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^{7/2} \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^{7/2} \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 {84 b^2 x^{5/2} \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 {28 b x^3 \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 {84 b^2 x^{5/2} \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 {28 b x^3 \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^{7/2} \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 (420 b^2\right ) \text {Subst}\left (\int x^4 \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 (420 b^2\right ) \text {Subst}\left (\int x^4 \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 {(168 b) \text {Subst}\left (\int x^5 \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 {(168 b) \text {Subst}\left (\int x^5 \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 (14 b^3\right ) \text {Subst}\left (\int x^6 \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 (14 b^3\right ) \text {Subst}\left (\int x^6 \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 = 10.37 (sec) , antiderivative size = 3035, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {x^{3}}{\left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]
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\[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]
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