Integrand size = 20, antiderivative size = 677 \[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {315 b^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}-\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )}{d^7}+\frac {315 b^2 \operatorname {PolyLog}\left (7,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d} \]
[Out]
Time = 0.59 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5544, 4275, 4265, 2611, 6744, 2320, 6724, 4269, 3799, 2221} \[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {315 b^2 \operatorname {PolyLog}\left (7,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {315 b^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {14 b^2 x^3 \log \left (e^{2 \left (c+d \sqrt {x}\right )}+1\right )}{d^2}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {2 b^2 x^{7/2}}{d} \]
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 4265
Rule 4269
Rule 4275
Rule 5544
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^7 (a+b \text {sech}(c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^7+2 a b x^7 \text {sech}(c+d x)+b^2 x^7 \text {sech}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}+(4 a b) \text {Subst}\left (\int x^7 \text {sech}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^7 \text {sech}^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(28 i a b) \text {Subst}\left (\int x^6 \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(28 i a b) \text {Subst}\left (\int x^6 \log \left (1+i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {\left (14 b^2\right ) \text {Subst}\left (\int x^6 \tanh (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(168 i a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(168 i a b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (28 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x^6}{1+e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(840 i a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(840 i a b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(3360 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(3360 i a b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (210 b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(10080 i a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(10080 i a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}-\frac {\left (420 b^2\right ) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(20160 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {(20160 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^6}+\frac {\left (630 b^2\right ) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {315 b^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )}{d^7}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(20160 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {(20160 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^7}-\frac {\left (630 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (5,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {315 b^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}-\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )}{d^7}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(20160 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^8}+\frac {(20160 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(7,i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^8}+\frac {\left (315 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (6,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {315 b^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}-\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {\left (315 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(6,-x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8} \\ & = \frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}+\frac {8 a b x^{7/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {14 b^2 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 i a b x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 i a b x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 i a b x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 i a b x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {315 b^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 i a b x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 i a b x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )}{d^6}-\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 i a b \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )}{d^7}+\frac {315 b^2 \operatorname {PolyLog}\left (7,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 i a b \operatorname {PolyLog}\left (8,-i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 i a b \operatorname {PolyLog}\left (8,i e^{c+d \sqrt {x}}\right )}{d^8}+\frac {2 b^2 x^{7/2} \tanh \left (c+d \sqrt {x}\right )}{d} \\ \end{align*}
Time = 8.60 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.10 \[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {\cosh \left (c+d \sqrt {x}\right ) \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \left (\frac {16 b^2 e^{2 c} x^{7/2} \cosh \left (c+d \sqrt {x}\right )}{d \left (1+e^{2 c}\right )}+a^2 x^4 \cosh \left (c+d \sqrt {x}\right )+\frac {2 i b \cosh \left (c+d \sqrt {x}\right ) \left (8 a d^7 x^{7/2} \log \left (1-i e^{c+d \sqrt {x}}\right )-8 a d^7 x^{7/2} \log \left (1+i e^{c+d \sqrt {x}}\right )+28 i b d^6 x^3 \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )-56 a d^6 x^3 \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )+56 a d^6 x^3 \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )+84 i b d^5 x^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )+336 a d^5 x^{5/2} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )-336 a d^5 x^{5/2} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )-210 i b d^4 x^2 \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )-1680 a d^4 x^2 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )+1680 a d^4 x^2 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )+420 i b d^3 x^{3/2} \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )+6720 a d^3 x^{3/2} \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )-6720 a d^3 x^{3/2} \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )-630 i b d^2 x \operatorname {PolyLog}\left (5,-e^{2 \left (c+d \sqrt {x}\right )}\right )-20160 a d^2 x \operatorname {PolyLog}\left (6,-i e^{c+d \sqrt {x}}\right )+20160 a d^2 x \operatorname {PolyLog}\left (6,i e^{c+d \sqrt {x}}\right )+630 i b d \sqrt {x} \operatorname {PolyLog}\left (6,-e^{2 \left (c+d \sqrt {x}\right )}\right )+40320 a d \sqrt {x} \operatorname {PolyLog}\left (7,-i e^{c+d \sqrt {x}}\right )-40320 a d \sqrt {x} \operatorname {PolyLog}\left (7,i e^{c+d \sqrt {x}}\right )-315 i b \operatorname {PolyLog}\left (7,-e^{2 \left (c+d \sqrt {x}\right )}\right )-40320 a \operatorname {PolyLog}\left (8,-i e^{c+d \sqrt {x}}\right )+40320 a \operatorname {PolyLog}\left (8,i e^{c+d \sqrt {x}}\right )\right )}{d^8}+\frac {8 b^2 x^{7/2} \text {sech}(c) \sinh \left (d \sqrt {x}\right )}{d}\right )}{4 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )^2} \]
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\[\int x^{3} \left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right )^{2}d x\]
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\[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
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\[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^3\,{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
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